Polar code rate matching method and device

Abstract

This application discloses a method, device and medium for Polar code rate matching in satellite communications, which can improve the reliability of the system. The method includes: obtaining K bits to be encoded, obtaining a mother code length N, N = 2 n Where K, n, and N are positive integers, polar coding is performed on the K bits to be coded according to the mother code length N, and rate matching is performed on the polar coded bit sequence according to a rate matching sequence before outputting the result. The given rate matching sequence can simplify the rate matching operation.

Description

Technical Field

[0001] The embodiments of the present application relate to the field of communication technology, and more particularly to a Polar code rate method and apparatus. Background Art

[0002] As the most basic wireless access technology, channel coding plays a vital role in ensuring reliable data transmission. In existing wireless communication systems, Turbo codes, low-density parity check (LDPC) codes, and polar codes are generally used for channel coding. Turbo codes cannot support information transmission at too low or too high code rates. Furthermore, for medium and short packet transmission, Turbo codes and LDPC codes, due to their inherent coding characteristics, find it difficult to achieve ideal performance with limited code lengths. In terms of implementation, Turbo codes and LDPC codes have high computational complexity during the coding implementation process. Polar codes are theoretically proven to achieve Shannon capacity and have relatively simple coding complexity, and therefore are gaining increasing popularity. In fifth-generation (5G) communication systems, Polar codes are used as the coding method for control channels.

[0003] However, in satellite communications, it is not yet clear what channel coding method to use. If Polar codes are to be introduced into satellite communication systems, what kind of design is needed to meet the requirements has become a problem that needs to be solved. Summary of the Invention

[0004] The embodiments of the present application provide a rate matching method and apparatus for Polar code encoding applied to satellite communications, which has the advantage of being simple to implement.

[0005] The specific technical solutions provided in the embodiments of this application are as follows:

[0006] In a first aspect, a rate matching method for Polar code encoding is provided.

[0007] Get K bits to be encoded, target code length M, mother code length N, N = 2 n , M, K, n and N are positive integers;

[0008] Performing polar coding on the K to-be-coded bits according to the mother code length N to obtain a coded bit sequence;

[0009] Performing rate matching on the encoded bit sequence according to a rate matching sequence to obtain a rate-matched sequence;

[0010] Output the rate-matched sequence.

[0011] This design is in line with the characteristics of satellite communications. The nested nature of the rate matching sequence also provides multiple possible ways for specific implementation, which is easy to implement, simplifies the rate matching operation, and reduces the difficulty of implementation.

[0012] In one possible design, the rate matching sequence is a subsequence of a first sequence, where the first sequence is sequence 1, sequence 2, or sequence 3 in Appendix 1, wherein sequence 1 includes N2 polarization subchannel numbers, sequence 2 includes N2 / 2 polarization subchannel numbers, and sequence 3 includes N2 values ​​of different sizes. The polarization subchannel numbers start from 0, and N2 is greater than or equal to N.

[0013] In one possible design, when N=8192, the rate matching sequence is sequence 1a, sequence 2a, or sequence 3a in the specification; or

[0014] When N=4096, the rate matching sequence is sequence 1b, 2b, or 3b in the specification; or

[0015] When N=2048, the rate matching sequence is sequence 1c, 2c, or 3c in the specification; or

[0016] When N=1024, the rate matching sequence is sequence 1d, 2d, or 3d in the specification; or

[0017] When N=512, the rate matching sequence is the sequence 1e, 2e, or 3e in the specification;

[0018] wherein the sequence 1a, the sequence 1b, the sequence 1c, the sequence 1d, and the sequence 1e are subsequences of the sequence 1;

[0019] Sequence 2a, sequence 2b, sequence 2c, sequence 2d, and sequence 2e are subsequences of sequence 2;

[0020] The sequence 3a, the sequence 3b, the sequence 3c, the sequence 3d, and the sequence 3e are subsequences of the sequence 3.

[0021] In one possible design, the rate matching method is a puncturing operation.

[0022] If the first sequence is sequence 1, the puncturing operation is: taking bits carried by subchannels corresponding to the last M polarized subchannel numbers in the rate matching sequence according to the target code length M, or deleting bits carried by subchannels corresponding to the first NM polarized subchannel numbers in the rate matching sequence according to the target code length M; or

[0023] If the first sequence is sequence 2, the puncturing operation is: deleting bits carried by subchannels corresponding to the first NM polarized subchannel numbers in the rate matching sequence according to the target code length M; or

[0024] If the first sequence is sequence 3, the puncturing operation is: deleting bits carried by the polarized subchannels containing the NM smallest values ​​in the rate matching sequence according to the target code length M, or extracting bits carried by the polarized subchannels containing the M largest values ​​in the rate matching sequence according to the target code length M.

[0025] In one possible design, polarization coding the K to-be-coded bits according to the mother code length N includes mapping the K to-be-coded bits to K polarization subchannels with higher reliability among MP non-punctured polarization subchannels, such that the reliability of the polarization subchannel containing the K to-be-coded bits is higher than the reliability of the remaining MPK polarization subchannels, where P is the number of pre-frozen polarization subchannels, and P is greater than or equal to 0.

[0026] In one possible design, before performing rate matching on the encoded bit sequence according to a rate matching sequence to obtain a rate-matched sequence, the encoded bit sequence is interleaved.

[0027] In one possible design, the K bits to be encoded include cyclic redundancy check bits.

[0028] In one possible design, the K bits to be encoded include parity bits.

[0029] In one possible design, the rate matching method is performed by a network device or a terminal.

[0030] In a second aspect, a Polar code rate matching apparatus is provided, which implements the method described in the first aspect and any possible design of the first aspect. The functionality can be implemented in hardware or by hardware executing corresponding software. The hardware or software includes one or more modules corresponding to the functionality described above.

[0031] In one possible design, when part or all of the functions are implemented through hardware, the Polar code rate matching device includes: an input interface circuit for obtaining bits to be encoded; a logic circuit for executing the actions described in the first aspect and any possible design of the first aspect; and an output interface circuit for outputting a rate-matched sequence.

[0032] Optionally, the Polar code rate matching device may be a chip or an integrated circuit.

[0033] In one possible design, when part or all of the functions are implemented by software, the Polar code rate matching device includes: a memory for storing a program; and a processor for executing the program stored in the memory. When the program is executed, the Polar code rate matching device can implement the method as described in the first aspect and any possible design of the first aspect above.

[0034] Optionally, the above-mentioned memory may be a physically independent unit or may be integrated with the processor.

[0035] In one possible design, when some or all of the functions are implemented via software, the Polar code rate matching device includes a processor. A memory for storing programs is located outside the rate matching device, and the processor is connected to the memory via circuits / wires to read and execute the programs stored in the memory.

[0036] In one possible design, the apparatus is a network device or a terminal.

[0037] In a third aspect, a communication system is provided, which includes a network device and a terminal, and the network device or the terminal can execute the method described in the first aspect and its possible designs.

[0038] According to a fourth aspect, a computer storage medium is provided, storing a computer program, wherein the computer program includes instructions for executing the method described in the first aspect and any one of any possible designs of the first aspect.

[0039] In a fifth aspect, an embodiment of the present application provides a computer program product comprising instructions, which, when executed on a computer, enables the computer to execute the methods described in the above aspects.

[0040] In a sixth aspect, a wireless device is provided, comprising a rate matching device and a transceiver for implementing the first aspect and any possible design of the first aspect,

[0041] The transceiver is used to receive or send signals.

[0042] In one possible design, the wireless device is a terminal or a network device. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 Schematic diagram of the communication system architecture used in the embodiments of the present application;

[0044] Figure 2 Schematic diagram of the process of the Polar code rate matching method in the embodiment of the present application;

[0045] Figure 3This is one of the structural diagrams of the Polar code rate matching device in the embodiment of this application;

[0046] Figure 4 This is the second structural diagram of the Polar code rate matching device in the embodiment of this application;

[0047] Figure 5 This is the third structural diagram of the Polar code rate matching device in the embodiment of this application;

[0048] Figure 6 This is the fourth structural diagram of the Polar code rate matching device in the embodiment of the present application;

[0049] Figure 7 This is one of the structural diagrams of the Polar code decoding rate matching device in the embodiment of this application;

[0050] Figure 8 This is the second structural diagram of the Polar code decoding rate matching device in the embodiment of the present application;

[0051] Figure 9 This is the third structural diagram of the Polar code decoding rate matching device in the embodiment of this application;

[0052] Figure 10 This is the fourth structural diagram of the Polar code decoding rate matching device in the embodiment of this application. DETAILED DESCRIPTION

[0053] The embodiments of the present application will be described in detail below with reference to the accompanying drawings.

[0054] Directly applying the Polar code encoding scheme from the current 5G standard to satellite communications without considering the specific characteristics of satellite communications would result in at least the following issues: 1) Polar codes in the current 5G standard are only used for control channels and therefore may not be fully applicable to data channels; 2) The maximum mother code length of Polar codes in the current 5G standard is 1024, which limits the supported data packet length. Therefore, it is necessary to redesign the Polar code encoding scheme to make it more suitable for satellite communications while also ensuring compatibility with terrestrial communications.

[0055] This embodiment of the present application provides a method and apparatus for rate matching of polar codes. Based on the reliability of polarized subchannels, a reliability ranking is calculated online or pre-stored. The sequence number of the polarized subchannel used to transmit information bits is selected based on the reliability ranking. Polar code encoding is performed according to the selected information bit sequence number. The encoded sequence is then rate-matched according to a rate matching sequence, and the rate-matched sequence is output. This embodiment of the present application is particularly suitable for satellite communications, particularly short message services in satellite communications. The rate matching method and apparatus provided by the embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0056] To facilitate understanding of the embodiments of the present application, a brief introduction to Polar codes is given below.

[0057] Polar code's coding strategy uses a noiseless channel to transmit useful information for users, and a full-noise channel to transmit agreed information or no information. Polar code is also a linear block code, and its coding matrix is ​​G N , the encoding process is in is a binary row vector with a length of N (i.e., code length. As you can see, the lengths of the sequences x and u before and after encoding are both N, also known as the mother code length); G N is an N×N matrix, and It is defined as the Kronecker product of log2N matrices F2. The above matrix

[0058] During the encoding process of Polar code, A part of the bits in is used to carry information, which is called the information bit set. The set of indexes of these bits is recorded as The other part of the bits is set to a fixed value agreed upon by the receiving end and the sending end, which is called the frozen bit set or frozen bit set (frozen bits). The set of its index is The complement of The encoding process of Polar code is equivalent to: Here, G N (A) is G N Zhongyou Collection The submatrix G is obtained by taking the rows corresponding to the indices in N (A C ) is G N Zhongyou Collection The submatrix obtained by indexing the rows corresponding to . for The number of information bits in the set is K. Generally, various check bits including but not limited to cyclic redundancy check (CRC) bits and parity check (PC) bits can also be included in the information bit set; for The set of frozen bits in , whose number is (NK), are known bits. These frozen bits are usually set to 0, but as long as the receiver and sender agree in advance, the frozen bits can be set arbitrarily. Therefore, the encoded output of the Polar code can be simplified to: here for The set of information bits in , is a row vector of length K, that is |·| represents the number of elements in the set, K is the information block size, is the matrix G N Zhongyou Collection The submatrix obtained by the rows corresponding to the indices in It is a K×N matrix.

[0059] After determining the mother code length N, the construction process of Polar code is to set The selection process determines the performance of Polar codes. The construction process of Polar codes is usually to determine the N polarization sub-channels that exist according to the mother code length N, corresponding to the N rows of the coding matrix. Without considering rate matching, the indexes of the first K polarization sub-channels with higher reliability are used as the set The indexes corresponding to the remaining (NK) polarization sub-channels are used as the index set of frozen bits. Elements of a set Determines the location of the information bit, the set Determines the position of the frozen bit. The serial number of the polarized subchannel is the position index of the information bit or frozen bit, that is, The position index in .

[0060] When considering rate matching, primarily puncturing or shortening, the first step is to determine the NM polarized subchannels to be punctured or shortened (i.e., deleted). Here, M is the target code length, i.e., the length of the bit sequence after rate matching. These NM polarized subchannels are used to place frozen bits. The 5G NR standard also defines some pre-frozen polarized subchannels for puncturing, also used to place frozen bits. Here, let's define the number of pre-frozen polarized subchannels as P, where P is greater than or equal to 0. (For shortening, P is obviously 0 and is not considered in this case.) Then, from the remaining MP polarized subchannels, K with higher reliability are selected based on their reliability to place the K information bits. Alternatively, MPK subchannels with lower reliability can be selected first to place frozen bits, while the remaining K subchannels are used to place information bits. The reliability of any of the K polarization subchannels for placing K information bits is higher than the reliability of any of the MPK subchannels for placing frozen bits. In this application, there is no restriction on the value of P. Even in the case of puncturing, P can be 0, and even in the case of shortening, P can be greater than 0. This does not affect the implementation of the technical solution of this application. Whether it is the encoding end or the decoding end, the principles and methods for confirming that K subchannels are used to place K information bits are consistent. At the same time, in the 5G NR standard, for the selection of NM polarization subchannels, the sequence after sub-block interleaving (divided into 32 sub-blocks) is placed in a circular buffer (equivalent to a rate matching sequence in English). If puncturing is performed, the circular buffer is read starting from the NMth position and bits from 0 to NM-1 are discarded. If shortening is performed, the circular buffer is read starting from the 0th position to the M-1th position and bits from M to N-1 are discarded. This approach does not consider the relationship between rate matching sequences of different mother code lengths.

[0061] It should be noted that the relative reliability relationship mentioned here is based on a given reliability calculation method. Different reliability calculation methods may cause the relative reliability relationship of the polarization subchannel to change, but the method of selecting the polarization subchannel for placing information bits is the same. Outside the 5G NR standard, information bits can also be considered to be placed in the polarization subchannel that is finally punctured or shortened. This application does not limit the selection of the polarization subchannel for placing information bits to the 5G NR standard.

[0062] Current satellite communications, such as the Beidou satellite communications system, utilize a short message communication mode. This mode can be considered the commonly used "short message" mode, but it differs in that the information published not only displays the information itself but also indicates both the location of the sender and the sender. However, the Turbo code currently used in the Beidou-2 short message system can be replaced with Polar code to improve performance and reduce complexity. When sending information bits to the ground center, Beidou short messages are first segmented. Each segment has a length of 89 to 994 bits (including CRC bits), using a code rate of 1 / 2. Subsequent consumer terminals that wish to access the short message system will need to use a lower code rate, such as 1 / 8.

[0063] In the embodiment of the present application, by pre-given rate matching sequences, the rate matching sequences corresponding to different mother code lengths also satisfy a nested relationship, which can simplify the operation process and improve processing efficiency. This nested feature is similar to the reliability sorting sequence corresponding to different mother code lengths in the 5G NR standard, that is, when the mother code length Na is less than or equal to the mother code length Nb, the relative reliability relationship of each polarization sub-channel (polarization sub-channel 0, ..., Na-1) in the reliability sequence corresponding to Na is consistent with the relative reliability relationship of these polarization sub-channels (polarization sub-channel 0, ..., Na-1) in the reliability sequence corresponding to Nb. In other words, in the reliability sequence corresponding to Nb, only the polarization sub-channel numbers less than Na are retained and the relative relationship between the positions of the Na polarization sub-channel numbers is not changed, so that the reliability sequence corresponding to Na can be obtained. Therefore, in the specific implementation, either all reliability sequences of different mother code lengths can be pre-stored, or only the maximum mother code length N can be pre-stored. max The corresponding reliability sequence, then the reliability sequences corresponding to other mother code lengths Na are only retained from the reliability sequences corresponding to the maximum mother code length. max This can save some storage overhead and also facilitate implementation without considering operations such as sub-block interleaving. In particular, Na can also be equal to N at most. max .

[0064] Similarly, when the mother code length Na is less than or equal to the mother code length Nb, the relative relationship of the rate matching (puncturing or shortening) order of each polarization subchannel (polarization subchannel 0, ..., Na-1) in the rate matching sequence corresponding to Na is consistent with the relative relationship of the rate matching order of these polarization subchannels (polarization subchannel 0, ..., Na-1) in the rate matching sequence corresponding to Nb. In other words, in the rate matching sequence corresponding to Nb, only the polarization subchannel numbers less than Na are retained and the relative relationship of the positions of the polarization subchannel numbers Na is not changed, so that the rate matching sequence corresponding to Na can be obtained. Therefore, in a specific implementation, either all rate matching sequences of different mother code lengths can be pre-stored, or only the rate matching sequence with the maximum mother code length N can be pre-stored. max The corresponding rate matching sequence, then the rate matching sequences corresponding to other mother code lengths Na are retained from the rate matching sequences corresponding to the maximum mother code length only those less than N max The number of the polarization subchannel is obtained, in particular, Na can also be equal to N max .

[0065] Figure 1 Schematic diagram of the structure of a wireless communication network according to an embodiment of the present invention. Figure 1 This is only an example, and other wireless networks that can use the rate matching method or device of the embodiment of the present invention are also within the protection scope of the present invention.

[0066] like Figure 1 As shown, the wireless communication network 100 includes a network device 110 and a terminal 112. When the wireless communication network 100 includes a core network 102, the network device 110 can also be connected to the core network 102. The network device 110 can also communicate with an IP network 104, such as the Internet, a private IP network, or other data networks. The network device provides services to terminals within the coverage area. For example, see Figure 1 As shown, network device 110 provides wireless access to one or more terminals 112 within the coverage area of ​​network device 110. In addition, the coverage areas of network devices may overlap, such as network devices 110 and 120. Network devices may also communicate with each other, such as network device 110 and network device 120.

[0067] The above-mentioned network device can be a device for communicating with a terminal device. For example, it can be an evolved Node B (eNB or eNodeB) in an LTE system, a gNB in ​​a 5G network, a satellite in satellite communications, or a network-side device in a future communication system. Alternatively, the network device can be a relay station, an access point, an on-board device, etc. In a device-to-device (D2D) communication system, a machine-to-machine (M2M) communication system, and a vehicle network system, the network device can also be a terminal that functions as a base station.

[0068] The term "terminal" may refer to user equipment (UE), access terminal, subscriber unit, mobile station, remote station, remote terminal, mobile device, user terminal, wireless communication device, user agent, or user device. An access terminal may be a cellular phone, a cordless phone, a Session Initiation Protocol (SIP) phone, a Wireless Local Loop (WLL) station, a Personal Digital Assistant (PDA), a handheld device with wireless communication capabilities, a computing device or other processing device connected to a wireless modem, an in-vehicle device, a wearable device, or a terminal device in a future communication network.

[0069] based on Figure 1 In the communication system architecture shown in the embodiment of the present application, the execution subject of the Polar code rate matching method can be the aforementioned network device or terminal. When the network device or terminal acts as a transmitter to send data or information, this Polar code rate matching method can be used. Correspondingly, when the network device or terminal acts as a receiver to receive data or information, it is also necessary to first confirm the rate matching sequence according to the method of the present invention. The Polar code rate matching method provided in the embodiment of the present application is described in detail below.

[0070] based on Figure 1 The communication system architecture shown in Figure 2 As shown, the specific process of the Polar code rate matching method provided in the embodiment of the present application is described as follows.

[0071] Operation 210: Obtain K bits to be encoded, target code length M, and mother code length N, where N=2 n , M, K, n and N are positive integers;

[0072] The K bits to be encoded herein may include, in addition to payload bits, one or more of various other types of non-frozen bits, including but not limited to CRC bits, PC bits, and the like.

[0073] For the sake of system simplicity, the present invention does not consider repeating this rate matching method. Therefore, the mother code length N can be obtained according to the target code length: N = 2^ceil(log2(M)), that is, n = ceil(log2(M)). The ceil operation here refers to a ceiling operation. It can be seen that N is the smallest number that is not less than M and satisfies the integer power of 2.

[0074] Furthermore, if a fixed bit rate is used, the target code length M can be calculated when K bits to be encoded are obtained, or the target code length M can be used to determine the number of bits to be encoded. For example, if a fixed bit rate of 1 / 8 is used, then M = 8K, or K = M / 8. Fixed bit rates can also be commonly used, such as 1 / 2, 1 / 3, or 1 / 4. This approach can further reduce implementation complexity. If a variable bit rate is used, signaling or other means are required to ensure consistency between the sender and receiver. This approach offers the advantage of more flexible resource utilization. In specific applications, this can be specified in advance through standard or proprietary protocols.

[0075] Operation 220: Perform polar coding on the K to-be-coded bits according to the mother code length N to obtain a coded bit sequence;

[0076] For specific coding operations, please refer to the above description. If an operation similar to the 5G NR standard is followed, the polarization subchannel to be punctured or shortened is first determined. In this step, a rate matching sequence is required to determine which polarization subchannels to puncture or shorten (i.e., delete) and the possible P pre-frozen polarization subchannels (P is greater than or equal to 0). Then, K polarization subchannels with higher reliability are selected from the remaining MP polarization subchannels according to their reliability to place the K information bits. The decoding side also determines the position of the K information bits in the same way. To avoid repetition, the description of the rate matching sequence is provided in operation 230. It should be noted that in the 5G NR system, it is necessary to confirm whether to use puncturing or shortening rate matching each time rate matching is performed. In this application, for the sake of simplicity, puncturing or shortening can be fixed as the rate matching method.

[0077] Optionally, the coded bit sequence is interleaved. This interleaving operation is slightly different from the sub-block interleaving in 5G NR. Instead of dividing the bit sequence into 32 sub-blocks (or any other number of sub-blocks), the coded bit sequence is mapped to a circular buffer according to the rate matching sequence for subsequent puncturing (of course, the shortening operation is similar).

[0078] Operation 230: performing a rate matching operation on the encoded bit sequence according to the rate matching sequence to obtain a rate-matched bit sequence of length M;

[0079] Appendix 1 provides an exemplary first sequence for puncturing operations. The first sequence is sequence 1, sequence 2, or sequence 3. Sequence 1 includes N2 polarization subchannel numbers, and sequence 2 includes N2 / 2 polarization subchannel numbers. The minimum value of a polarization subchannel number is 0, and the maximum value is N2-1, where N2 = 8192, and N2 is greater than or equal to N. Alternatively, the minimum value of a polarization subchannel number can be 1, and the maximum value can be N2. In this case, all polarization subchannel numbers involved are incremented by 1. The first sequence can also be sequence 3, which includes N2 values ​​of varying sizes. The sequence number of each value is the polarization subchannel number, and the size of the value represents the rate matching priority of the polarization subchannel corresponding to that value.

[0080] It should be noted that sequences 1, 2, and 3 are essentially equivalent. For the puncturing operation, sequence 2 is essentially a subsequence of sequence 1, directly taking the first half of sequence 1, and the value of each element in sequence 3 represents the puncturing priority of the position where the element is located (i.e., the polarization subchannel). The smaller the value, the higher the puncturing priority. According to sequence 1 or sequence 2, if the target code length is M, then the bits carried on the polarization subchannel corresponding to the first NM polarization subchannel numbers in sequence 1 or sequence 2 need to be punctured (or deleted). For sequence 3, the bits carried on the polarization subchannel with the smallest NM values ​​in sequence 3 need to be punctured, or the bits carried on the polarization subchannel with the largest M values ​​are retained.

[0081] Communication systems usually define a maximum mother code length N max , if N max= 8, sequence 1 is [0, 4, 6, 2, 3, 1, 7, 5]. Correspondingly, sequence 2 is [0, 4, 6, 2], which is the first half of sequence 1, and sequence 3 is [0, 5, 3, 4, 1, 7, 2, 6]. The '5' in sequence 1 represents polarization subchannel 5, which has a puncturing priority of 7 and is the last to be punctured. This corresponds to the '7' in sequence 3, meaning that polarization subchannel 5 (the fifth element from element 0) in sequence 3 has a puncturing priority of '7'. If M = 6, then in sequence 1 or sequence 2, the bits carried by the first NM = 2 polarization subchannels (polarization subchannels 0 and 4) in the sequence are punctured. Therefore, the bits punctured are the polarization subchannels with values ​​less than or equal to NM = 2 (i.e., 0 and 1) (polarization subchannels 0 and 4). According to the nested characteristics, when N = 4, the sequence corresponding to sequence 1 is [0, 2, 3, 1], and the sequence corresponding to sequence 2 is [0, 2]. Note that in sequence 3, since the value only represents the priority, the value is not necessarily limited to the largest polarization sub-channel number. Under different value selection methods, it is only necessary to confirm that the priority order of different polarization sub-channels is consistent. Therefore, when N = 4, for sequence 3, we can simply take N max = 8, that is, the sequence corresponding to sequence 3 is [0, 5, 3, 4], where 5 is already larger than N. However, this method is equivalent to the sequence 3 (i.e., [0, 3, 1, 2]) derived from sequence 1 (i.e., [0, 2, 3, 1]) when N = 4. That is, when the rate matching method is puncturing, both methods indicate that the priority of the 0th polarization subchannel to be punctured is 0th (i.e., the highest), the priority of the 1st polarization subchannel to be punctured is 3rd (i.e., the lowest), the priority of the 2nd polarization subchannel to be punctured is 1st, and the priority of the 3rd polarization subchannel to be punctured is 1st. The priority is 2nd. Similarly, when the rate matching mode is shortening, both modes indicate that the priority of shortening the 0th polarization subchannel is 0th (i.e., the highest), the priority of shortening the 1st polarization subchannel is 3rd (i.e., the highest), the priority of shortening the 2nd polarization subchannel is 1st, and the priority of shortening the 3rd polarization subchannel is 2nd. Therefore, the nesting mode of sequence 3 can be similar to the nesting mode of sequence 1, on the one hand, selecting a value less than or equal to N from a sequence greater than N, and on the other hand, it can be more simply to take the values ​​of the first N positions from the third sequence with a length greater than N. In fact, since sequences 1, 2, and 3 all meet the nesting characteristics, and sequence 2 is the first half of sequence 1, and sequence 3 is the expression of sequence 1 in the form of puncturing priority, not all sequences can meet this requirement. N is still used. max Taking N=8 as an example, if sequence 1 is [0, 2, 4, 3, 6, 1, 7, 5], it cannot satisfy the nesting of sequence 2 when N=4. Therefore, when N2 is large, the design of nested rate matching sequences is very difficult.

[0082] When the rate matching mode is shortening, Sequence 1 and Sequence 3 can be the same as Sequence 1 and Sequence 3 used in the puncturing operation (thus, Sequence 1 and Sequence 3 in Appendix 1 can be used directly), while Sequence 2 can be the second half of Sequence 1. They also satisfy the nesting property. The value of each element in Sequence 3 indicates the shortening priority of the location (i.e., polarization subchannel) where the element resides; the larger the value, the higher the shortening priority. For Sequence 1 or Sequence 2, if the target code length is M, the bits carried on the polarization subchannels corresponding to the last NM polarization subchannel numbers in Sequence 1 or Sequence 2 need to be shortened. For Sequence 3, the bits carried on the polarization subchannels with values ​​greater than or equal to M in Sequence 3 need to be shortened. If the above sequence [0, 4, 6, 2, 3, 1, 7, 5] is used as sequence 1, the corresponding sequence 2 is [3, 1, 7, 5]. When performing the shortening operation, what needs to be shortened are the bits carried on the polarization sub-channels corresponding to the last NM polarization sub-channel numbers of sequence 1 or sequence 2. For sequence 3, what needs to be shortened is the bits carried by the polarization sub-channels where the NM values ​​with the largest values ​​in sequence 3 are located. In other words, the bits carried by the polarization sub-channels where the M values ​​with the smallest values ​​in sequence 3 are located are retained and shortened. To avoid confusion, the rest of this application uses puncturing as an example, but the overall principle also applies to the shortening operation and will not be repeated here.

[0083] In practical applications, the rate matching sequence can be a subsequence of the first sequence. Note that in this application, a subsequence of a sequence can also include the sequence itself. Sequence A being a subsequence of sequence B means that all elements in sequence A are taken from sequence B, and the relative order of these elements is exactly the same as their relative order in sequence B. When N = 8192, the rate matching sequence is 1a, 2a, or 3a. Sequence 1a is identical to sequence 1, sequence 2a is identical to sequence 2, and sequence 3a is identical to sequence 3. Therefore, sequences 1a, 2a, and 3a are not repeated. Appendix 1 also provides rate matching sequences 1b, 2b, and 3b with a mother code length of N=4096 (sequence 2b is 2048 long), rate matching sequences 1c, 2c, and 3c with a mother code length of N=2048 (sequence 2c is 1024 long), rate matching sequences 1d, 2d, and 3d with a mother code length of 1024 (sequence 2d is 512 long), and rate matching sequences 1e, 2e, and 3e with a mother code length of 512 (sequence 2e is 256 long). Among them, sequence 1a, sequence 1b, sequence 1c, sequence 1d, and sequence 1e are subsequences of sequence 1 and all satisfy the nesting property described above; sequence 2a, sequence 2b, sequence 2c, sequence 2d, and sequence 2e are subsequences of sequence 2 and also satisfy the nesting property described above; sequence 3a, sequence 3b, sequence 3c, sequence 3d, and sequence 3e are subsequences of sequence 3, and sequence 3b is the first half of sequence 3a, sequence 3c is the first half of sequence 3b (i.e., the first 1 / 4 of sequence 3a), sequence 3d is the first half of sequence 3c (i.e., the first 1 / 8 of sequence 3a), and sequence 3e is the first half of sequence 3d (i.e., the first 1 / 16 of sequence 3a), all of which also satisfy the nesting property described above.

[0084] Generally, the communication system defines a maximum mother code length N. max , generally, N max Less than or equal to N2, that is, N<=N max <=N2. The sender and receiver can only store N max The corresponding puncturing sequence (i.e., rate matching sequence) can also store less than or equal to N max The puncturing sequence corresponding to the different mother code lengths can also store N max And the puncturing sequences corresponding to several commonly used mother code lengths.

[0085] In addition to the examples in Appendix 1, Appendix 2 gives another example of the first sequence. Accordingly, there are two possible sets of rate matching sequences available. In actual applications, one of them can be used. Unless otherwise specified, this application takes Appendix 1 as an example.

[0086] Operation 240: Output the rate-matched sequence.

[0087] Note that if modulation is considered, the output here should be a sequence modulated based on the rate-matched bit sequence, which is a symbol sequence. In this case, the length of the sequence is not necessarily M. If modulation is not considered, the rate-matched sequence is the rate-matched bit sequence. Since modulation itself is not an essential technology for implementing the embodiment of the present application, without affecting understanding, the two terms "rate-matched sequence" and "rate-matched bit sequence" are not strictly distinguished in this application.

[0088] above Figure 2 The illustrated embodiment is the operation of the transmitting end, and the operation of the receiving end is similar. After receiving the information sequence to be decoded, the receiver (or receiving device) also needs to use a similar method. Specifically, it obtains the information sequence to be decoded including K bits to be decoded, the target code length M, and the mother code length N, determines the rate matching sequence, and performs a rate matching operation and a decoding and demodulation operation on the information sequence to be decoded, thereby obtaining and outputting a decoding result, wherein the rate matching is to restore the punctured position or the shortened position. For the punctured position, the likelihood ratio (LLR) can be considered to be 0 during decoding, and for the shortened position, the LLR can be considered to be infinite during decoding. The detailed decoding algorithm is not limited here.

[0089] It should be noted that the first sequences in Appendix 1 and Appendix 2 are two preferred examples. Their application in the Polar encoding process will help improve the Polar code encoding and decoding performance. In any of the example sequences, without affecting the overall effect, adjustments or equivalent replacements can be made, including but not limited to the following aspects:

[0090] 1. Swap the positions of a few elements in a sequence. For example, the position of a sequence number can be adjusted within a set range. For example, if the range is set to 5, the position of the element with sequence number 10 can be adjusted within 5 positions to the left or right.

[0091] 2. Some elements in the sequence are adjusted, but the set of channels selected according to the sequence for transmitting T bits of information is consistent or similar.

[0092] 3. The sequence contains N elements starting from 0 and ending at N-1. The N elements starting from 0 and ending at N-1 represent the sequence numbers of the N polarization subchannels. In fact, the sequence numbers of the N polarization subchannels can also start from 1 and end at N. Simply add 1 to each sequence number in the above sequence. This is also the sequence number format used in the above calculation methods. Of course, other methods can also be used to represent the sequence numbers or identifiers of the above polarization subchannels. This specific expression does not affect the specific position of the polarization subchannel represented in the sequence. In particular, as mentioned above, the elements in sequences 3b, 3c, 3d, and 3e may not all be between 0 and N-1, but this does not affect the priority order of each polarization subchannel.

[0093] 4. The sequence numbers of the N polarized subchannels in the above sequence are arranged from smallest to largest according to their rate matching priorities. In practice, the sequence numbers of the N polarized subchannels can also be arranged from largest to smallest according to their rate matching priorities by reversing the order of the elements in the above sequence.

[0094] like Figure 3 As shown, the embodiment of the present application further provides a Polar code rate matching device 300 or a Polar code decoding rate matching device 300. Figure 3 Part or all of the Polar code rate matching method shown can be implemented by hardware or software.

[0095] For the Polar code rate matching device 300, based on Figure 2 The same inventive concept of the Polar code rate matching method shown in FIG. Figure 2 When implemented in hardware, the Polar code rate matching device 300 includes: an input interface circuit 301 for obtaining bits to be encoded; a logic circuit 302 for executing the above Figure 2 The Polar code rate matching method shown is described in detail in the previous method embodiments and will not be repeated here. Output interface circuit 303 is used to output the rate-matched sequence. Furthermore, the rate-matched sequence is also a modulated sequence, a modulated symbol sequence, which is output to transceiver 320. Transceiver 320 performs appropriate processing (including but not limited to digital-to-analog conversion and / or frequency conversion) on the modulated sequence before transmitting it via antenna 330. Optionally, Polar code rate matching device 300 can be implemented as a chip or integrated circuit.

[0096] Optionally, when part or all of the Polar code rate matching method in the above embodiment is implemented by software, such as Figure 4As shown, the Polar code rate matching apparatus 400 includes: a memory 401 for storing a program; and a processor 402 for executing the program stored in the memory 401. When the program is executed, the Polar code rate matching apparatus 400 can implement the Polar code rate matching method provided in the above embodiment.

[0097] Optionally, the memory 401 may be a physically independent unit or a Figure 5 As shown, the memory 501 and the processor 502 are integrated together.

[0098] Optionally, when part or all of the rate matching method in the above embodiment is implemented via software, the Polar code rate matching apparatus 400 may include only the processor 402. The memory 401 for storing programs is located outside the Polar code rate matching apparatus 400. The processor 402 is connected to the memory 401 via circuits / wires to read and execute the programs stored in the memory 401.

[0099] based on Figure 2 The Polar code rate matching method shown in Figure 6 As shown, the embodiment of the present application further provides a Polar code rate matching device 600, which is used to perform Figure 2 The Polar code rate matching method shown in FIG. 6 includes a Polar code rate matching apparatus 600 comprising:

[0100] An acquisition unit 601 is configured to acquire K bits to be encoded and a target code length M;

[0101] The encoding unit 602 is configured to determine a mother code length N, and perform polar coding on the K bits to be encoded according to the mother code length N to obtain an encoded bit sequence. For the specific encoding process, refer to operation 220 of the aforementioned embodiment.

[0102] The rate matching unit 603 is used to perform a rate matching operation on the encoded bit sequence according to the rate matching sequence to obtain a rate-matched bit sequence of length M (or a sequence that further modulates the bit sequence, also called a rate-matched sequence). For the specific rate matching method, please refer to operation 230 of the aforementioned embodiment.

[0103] Corresponding to the transmitting end, the device at the receiving end can also be designed similarly.

[0104] Polar code rate matching device 700 includes an input interface circuit 701 for acquiring the information sequence to be decoded, a logic circuit 702 for executing the aforementioned Polar code rate matching method to obtain decoded information bits, and an output interface circuit 703 for outputting the decoded information bits. Polar code rate matching device 700 may also include a transceiver 720 for acquiring the information sequence to be decoded via an antenna 730. This may be implemented as a chip or integrated circuit.

[0105] Optionally, when part or all of the Polar code decoding rate matching method in the above embodiment is implemented by software, such as Figure 8 As shown, the Polar code rate matching apparatus 800 includes: a memory 801 for storing a program; and a processor 802 for executing the program stored in the memory 801. When the program is executed, the Polar code rate matching apparatus 800 can implement the Polar code rate matching method provided in the above embodiment.

[0106] Optionally, the memory 801 may be a physically independent unit or a Figure 9 As shown, the memory 901 and the processor 902 are integrated together.

[0107] Optionally, when part or all of the rate matching method in the above embodiment is implemented via software, the Polar code rate matching apparatus 800 may include only the processor 802. The memory 801 for storing programs is located outside the Polar code rate matching apparatus 800. The processor 802 is connected to the memory 801 via circuits / wires and is configured to read and execute the programs stored in the memory 801.

[0108] The processor 402 and / or the processor 802 may be a central processing unit (CPU), a network processor (NP), or a combination of a CPU and an NP.

[0109] Processor 402 and / or processor 802 may further include a hardware chip. The hardware chip may be an application-specific integrated circuit (ASIC), a programmable logic device (PLD), or a combination thereof. The PLD may be a complex programmable logic device (CPLD), a field-programmable gate array (FPGA), a generic array logic (GAL), or any combination thereof.

[0110] The memory in the above embodiments may include a volatile memory, such as a random-access memory (RAM); the memory may also include a non-volatile memory, such as a flash memory, a hard disk drive (HDD) or a solid-state drive (SSD); the memory may also include a combination of the above types of memory.

[0111] Based on the above-mentioned Polar code decoding rate matching method, such as Figure 10 As shown, the embodiment of the present application further provides a Polar code rate matching device 1000. The Polar code rate matching device 1000 is used to perform the aforementioned Polar code decoding rate matching method. The Polar code decoding rate matching device 1000 includes:

[0112] An acquisition unit 1001 is configured to acquire an information sequence to be decoded;

[0113] The processing unit 1002 is configured to determine a mother code length N, and perform rate matching and decoding and demodulation operations on the information sequence to be decoded according to the mother code length N and the rate matching sequence, thereby obtaining and outputting a decoding result.

[0114] An embodiment of the present application further provides a computer storage medium storing computer program instructions. When the computer program instructions are executed by a computer, the aforementioned Polar code rate matching method or Polar code decoding rate matching method is executed.

[0115] An embodiment of the present application also provides a computer program product comprising instructions, which, when executed on a computer, enables the aforementioned Polar code rate matching method or Polar code decoding rate matching method to be executed.

[0116] Those skilled in the art will appreciate that the embodiments of the present application can be provided as methods, systems, or computer program products. Therefore, the present application can adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment in combination with software and hardware. Moreover, the present application can adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to magnetic disk storage, CD-ROM, optical storage, etc.) that contain computer-usable program code.

[0117] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the steps in the process. Figure 1 a process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0118] These computer program instructions may also be stored in a computer readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 a process or multiple processes and / or boxes Figure 1 The function specified in one or more boxes.

[0119] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operational steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing the instructions executed on the computer or other programmable device for implementing the process. Figure 1 a process or multiple processes and / or boxes Figure 1 A step that specifies a function in one or more boxes.

[0120] Although the preferred embodiments of the present application have been described, those skilled in the art may make additional changes and modifications to these embodiments once they have learned the basic creative concept. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present application.

[0121] Obviously, those skilled in the art may make various changes and modifications to the embodiments of the present application without departing from the spirit and scope of the embodiments of the present application. Thus, if these modifications and variations of the embodiments of the present application fall within the scope of the claims of the present application and their equivalents, the present application also intends to include such modifications and variations.

[0122] Appendix 1: An example of the first sequence:

[0123] Sequence 1:

[0124] Punc_Q=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83 ,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118, 119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,1 50,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,1 81,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,21 2,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243 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062,8063,8064,8065,8066,8067,8068,8069,8070,8071,8072,8073,8074,8075,8076,8077,8078,8079,8080,8081,8082,8083,8084,8085,8086,8087,8088,8089,8090,8091,8092,8093,8094,8095,8096,8097,8098,8099,8100,8101,8102,8103,8104,810 5,8106,8107,8108,8109,8110,8111,8112,8113,8114,8115,8116,8117,8118,8119,8120,8121,8122,8123,8124,8125,8126,8127,8128,8129,8130,8131,8132,8133,8134,8135,8136,8137,8138,8139,8140,8141,8142,8143,8144,8145,8146,8147,8148 ,8149,8150,8151,8152,8153,8154,8155,8156,8157,8158,8159,8160,8161,8162,8163,8164,8165,8166,8167,8168,8169,8170,8171,8172,8173,8174,8175,8176,8177,8178,8179,8180,8181,8182,8183,8184,8185,8186,8187,8188,8189,8190,8191].

[0125] Sequence 2:

[0126] Punc_Q_half=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,2048,2049,2050,2051,2052,2053,2054,2055,2056,2057,2058,2059,2060,2061,2062,2063,2064,2065,2066,2067,2068,2069,2070,2071,2072,2073,2074,2075,2076,2077,2078,2079,2080,2081,2082,2083,2084,2085,2086,2087,2088,2089,2090,2091,2092,2093,2094,2095,2096,2097,2098,2099,2100,2101,2102,2103,2104,2105,2106,2107,2108,2109,2110,2111,2112,2113,2114,2115,2116,2117,2118,2119,2120,2121,2122,2123,2124,2125,2126,2127,2128,2129,2130,2131,2132,2133,2134,2135,2136,2137,2138,2139,2140,2141,2142,2143,2144,2145,2146,2147,2148,2149,2150,2151,2152,2153,2154,2155,2156,2157,2158,2159,2160,2161,2162,2163,2164,2165,2166,2167,2168,2169,2170,2171,2172,2173,2174,2175,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,4096,4097,4098,4099,4100,4101,4102,4103,4104,4105,4106,4107,4108,4109,4110,4111,4112,4113,4114,4115,4116,4117,4118,4119,4120,4121,4122,4123,4124,4125,4126,4127,4128,4129,4130,4131,4132,4133,4134,4135,4136,4137,4138,4139,4140,4141,4142,4143,4144,4145,4146,4147,4148,4149,4150,4151,4152,4153,4154,4155,4156,4157,4158,4159,4160,4161,4162,4163,4164,4165,4166,4167,4168,4169,4170,4171,4172,4173,4174,4175,4176,4177,4178,4179,4180,4181,4182,4183,4184,4185,4186,4187,4188,4189,4190,4191,4192,4193,4194,4195,4196,4197,4198,4199,4200,4201,4202,4203,4204,4205,4206,4207,4208,4209,4210,4211,4212,4213,4214,4215,4216,4217,4218,4219,4220,4221,4222,4223,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,7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4825,4826,4827,4828,4829,4830,4831,4832,4833,4834,4835,4836,4837,4838,4839,4840,4841,4842,4843,4844,4845,4846,4847,4848,4849,4850,4851,4852,4853,4854,4855,4856,4857,4858,4859,4860,4861,4862,4863].

[0127] Sequence 3:

[0128] Punc_Z=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83 ,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118, 119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,1 50,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,1 81,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,21 2,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243 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062,8063,8064,8065,8066,8067,8068,8069,8070,8071,8072,8073,8074,8075,8076,8077,8078,8079,8080,8081,8082,8083,8084,8085,8086,8087,8088,8089,8090,8091,8092,8093,8094,8095,8096,8097,8098,8099,8100,8101,8102,8103,8104,810 5,8106,8107,8108,8109,8110,8111,8112,8113,8114,8115,8116,8117,8118,8119,8120,8121,8122,8123,8124,8125,8126,8127,8128,8129,8130,8131,8132,8133,8134,8135,8136,8137,8138,8139,8140,8141,8142,8143,8144,8145,8146,8147,8148 ,8149,8150,8151,8152,8153,8154,8155,8156,8157,8158,8159,8160,8161,8162,8163,8164,8165,8166,8167,8168,8169,8170,8171,8172,8173,8174,8175,8176,8177,8178,8179,8180,8181,8182,8183,8184,8185,8186,8187,8188,8189,8190,8191].

[0129] When N=8192, sequence 1a is sequence 1, sequence 2a is sequence 2, and sequence 3a is sequence 3;

[0130] When N=4096,

[0131] Sequence 1b is:

[0132] Punc_Q=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83 ,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118, 119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,1 50,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,1 81,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,21 2,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243 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4057,4058,4059,4060,4061,4062,4063,4064,4065,4066,4067,4068,4069,4070,4071,4072,4073,4074,4075,4076,4077,4078,4079,4080,4081,4082,4083,4084,4085,4086,4087,4088,4089,4090,4091,4092,4093,4094,4095].

[0133] Sequence 2b is:

[0134] Punc_Q_half=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,2048,2049,2050,2051,2052,2053,2054,2055,2056,2057,2058,2059,2060,2061,2062,2063,2064,2065,2066,2067,2068,2069,2070,2071,2072,2073,2074,2075,2076,2077,2078,2079,2080,2081,2082,2083,2084,2085,2086,2087,2088,2089,2090,2091,2092,2093,2094,2095,2096,2097,2098,2099,2100,2101,2102,2103,2104,2105,2106,2107,2108,2109,2110,2111,2112,2113,2114,2115,2116,2117,2118,2119,2120,2121,2122,2123,2124,2125,2126,2127,2128,2129,2130,2131,2132,2133,2134,2135,2136,2137,2138,2139,2140,2141,2142,2143,2144,2145,2146,2147,2148,2149,2150,2151,2152,2153,2154,2155,2156,2157,2158,2159,2160,2161,2162,2163,2164,2165,2166,2167,2168,2169,2170,2171,2172,2173,2174,2175,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,2176,2177,2178,2179,2180,2181,2182,2183,2184,2185,2186,2187,2188,2189,2190,2191,2192,2193,2194,2195,2196,2197,2198,2199,2200,2201,2202,2203,2204,2205,2206,2207,2208,2209,2210,2211,2212,2213,2214,2215,2216,2217,2218,2219,2220,2221,2222,2223,2224,2225,2226,2227,2228,2229,2230,2231,2232,2233,2234,2235,2236,2237,2238,2239,2240,2241,2242,2243,2244,2245,2246,2247,2248,2249,2250,2251,2252,2253,2254,2255,2256,2257,2258,2259,2260,2261,2262,2263,2264,2265,2266,2267,2268,2269,2270,2271,2272,2273,2274,2275,2276,2277,2278,2279,2280,2281,2282,2283,2284,2285,2286,2287,2288,2289,2290,2291,2292,2293,2294,2295,2296,2297,2298,2299,2300,2301,2302,2303,1536,1537,1538,1539,1540,1541,1542,1543,1544,1545,1546,1547,1548,1549,1550,1551,1552,1553,1554,1555,1556,1557,1558,1559,1560,1561,1562,1563,1564,1565,1566,1567,1568,1569,1570,1571,1572,1573,1574,1575,1576,1577,1578,1579,1580,1581,1582,1583,1584,1585,1586,1587,1588,1589,1590,1591,1592,1593,1594,1595,1596,1597,1598,1599,1600,1601,1602,1603,1604,1605,1606,1607,1608,1609,1610,1611,1612,1613,1614,1615,1616,1617,1618,1619,1620,1621,1622,1623,1624,1625,1626,1627,1628,1629,1630,1631,1632,1633,1634,1635,1636,1637,1638,1639,1640,1641,1642,1643,1644,1645,1646,1647,1648,1649,1650,1651,1652,1653,1654,1655,1656,1657,1658,1659,1660,1661,1662,1663,2304,2305,2306,2307,2308,2309,2310,2311,2312,2313,2314,2315,2316,2317,2318,2319,2320,2321,2322,2323,2324,2325,2326,2327,2328,2329,2330,2331,2332,2333,2334,2335,2336,2337,2338,2339,2340,2341,2342,2343,2344,2345,2346,2347,2348,2349,2350,2351,2352,2353,2354,2355,2356,2357,2358,2359,2360,2361,2362,2363,2364,2365,2366,2367,2368,2369,2370,2371,2372,2373,2374,2375,2376,2377,2378,2379,2380,2381,2382,2383,2384,2385,2386,2387,2388,2389,2390,2391,2392,2393,2394,2395,2396,2397,2398,2399,2400,2401,2402,2403,2404,2405,2406,2407,2408,2409,2410,2411,2412,2413,2414,2415,2416,2417,2418,2419,2420,2421,2422,2423,2424,2425,2426,2427,2428,2429,2430,2431,2560,2561,2562,2563,2564,2565,2566,2567,2568,2569,2570,2571,2572,2573,2574,2575,2576,2577,2578,2579,2580,2581,2582,2583,2584,2585,2586,2587,2588,2589,2590,2591,2592,2593,2594,2595,2596,2597,2598,2599,2600,2601,2602,2603,2604,2605,2606,2607,2608,2609,2610,2611,2612,2613,2614,2615,2616,2617,2618,2619,2620,2621,2622,2623,2624,2625,2626,2627,2628,2629,2630,2631,2632,2633,2634,2635,2636,2637,2638,2639,2640,2641,2642,2643,2644,2645,2646,2647,2648,2649,2650,2651,2652,2653,2654,2655,2656,2657,2658,2659,2660,2661,2662,2663,2664,2665,2666,2667,2668,2669,2670,2671,2672,2673,2674,2675,2676,2677,2678,2679,2680,2681,2682,2683,2684,2685,2686,2687,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023].

[0135] Sequence 3b is:

[0136] Punc_Z=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83 ,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118, 119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,1 50,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,1 81,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,21 2,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243 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862,1863,1864,1865,1866,1867,1868,1869,1870,1871,1872,1873,1874,1875,1876,1877,1878,1879,1880,1881,1882,1883,1884,1885,1886,1887,1888,1889,1890,1891,1892,1893,1894,1895,1896,1897,1898,1899,1900,1901,1902,1903,1904,1905,1906,1907,1908,1909,1910,1911,1912,1913,1914,1915,1916,1917,1918,1919,3072,3073,3074,3075,3076,3077,3078,3079,3080,3081,3082,3083,3084,3085,3086,3087,3088,3089,3090,3091,3092,3093,3094,3095,3096,3097,3098,3099,3100,3101,3102,3103,3104,3105,3106,3107,3108,3109,3110,3111,3112,3113,3114,3115,3116,3117,3118,3119,3120,3121,3122,3123,3124,3125,3126,3127,3128,3129,3130,3131,3132,3133,3134,3135,3136,3137,3138,3139,3140,3141,3142,3143,3144,3145,3146,3147,3148,3149,3150,3151,3152,3153,3154,3155,3156,3157,3158,3159,3160,3161,3162,3163,3164,3165,3166,3167,3168,3169,3170,3171,3172,3173,3174,3175,3176,3177,3178,3179,3180,3181,3182,3183,3184,3185,3186,3187,3188,3189,3190,3191,3192,3193,3194,3195,3196,3197,3198,3199,2048,2049,2050,2051,2052,2053,2054,2055,2056,2057,2058,2059,2060,2061,2062,2063,2064,2065,2066,2067,2068,2069,2070,2071,2072,2073,2074,2075,2076,2077,2078,2079,2080,2081,2082,2083,2084,2085,2086,2087,2088,2089,2090,2091,2092,2093,2094,2095,2096,2097,2098,2099,2100,2101,2102,2103,2104,2105,2106,2107,2108,2109,2110,2111,2112,2113,2114,2115,2116,2117,2118,2119,2120,2121,2122,2123,2124,2125,2126,2127,2128,2129,2130,2131,2132,2133,2134,2135,2136,2137,2138,2139,2140,2141,2142,2143,2144,2145,2146,2147,2148,2149,2150,2151,2152,2153,2154,2155,2156,2157,2158,2159,2160,2161,2162,2163,2164,2165,2166,2167,2168,2169,2170,2171,2172,2173,2174,2175,3456,3457,3458,3459,3460,3461,3462,3463,3464,3465,3466,3467,3468,3469,3470,3471,3472,3473,3474,3475,3476,3477,3478,3479,3480,3481,3482,3483,3484,3485,3486,3487,3488,3489,3490,3491,3492,3493,3494,3495,3496,3497,3498,3499,3500,3501,3502,3503,3504,3505,3506,3507,3508,3509,3510,3511,3512,3513,3514,3515,3516,3517,3518,3519,3520,3521,3522,3523,3524,3525,3526,3527,3528,3529,3530,3531,3532,3533,3534,3535,3536,3537,3538,3539,3540,3541,3542,3543,3544,3545,3546,3547,3548,3549,3550,3551,3552,3553,3554,3555,3556,3557,3558,3559,3560,3561,3562,3563,3564,3565,3566,3567,3568,3569,3570,3571,3572,3573,3574,3575,3576,3577,3578,3579,3580,3581,3582,3583,3712,3713,3714,3715,3716,3717,3718,3719,3720,3721,3722,3723,3724,3725,3726,3727,3728,3729,3730,3731,3732,3733,3734,3735,3736,3737,3738,3739,3740,3741,3742,3743,3744,3745,3746,3747,3748,3749,3750,3751,3752,3753,3754,3755,3756,3757,3758,3759,3760,3761,3762,3763,3764,3765,3766,3767,3768,3769,3770,3771,3772,3773,3774,3775,3776,3777,3778,3779,3780,3781,3782,3783,3784,3785,3786,3787,3788,3789,3790,3791,3792,3793,3794,3795,3796,3797,3798,3799,3800,3801,3802,3803,3804,3805,3806,3807,3808,3809,3810,3811,3812,3813,3814,3815,3816,3817,3818,3819,3820,3821,3822,3823,3824,3825,3826,3827,3828,3829,3830,3831,3832,3833,3834,3835,3836,3837,3838,3839,5248,5249,5250,5251,5252,5253,5254,5255,5256,5257,5258,5259,5260,5261,5262,5263,5264,5265,5266,5267,5268,5269,5270,5271,5272,5273,5274,5275,5276,5277,5278,5279,5280,5281,5282,5283,5284,5285,5286,5287,5288,5289,5290,5291,5292,5293,5294,5295,5296,5297,5298,5299,5300,5301,5302,5303,5304,5305,5306,5307,5308,5309,5310,5311,5312,5313,5314,5315,5316,5317,5318,5319,5320,5321,5322,5323,5324,5325,5326,5327,5328,5329,5330,5331,5332,5333,5334,5335,5336,5337,5338,5339,5340,5341,5342,5343,5344,5345,5346,5347,5348,5349,5350,5351,5352,5353,5354,5355,5356,5357,5358,5359,5360,5361,5362,5363,5364,5365,5366,5367,5368,5369,5370,5371,5372,5373,5374,5375,2432,2433,2434,2435,2436,2437,2438,2439,2440,2441,2442,2443,2444,2445,2446,2447,2448,2449,2450,2451,2452,2453,2454,2455,2456,2457,2458,2459,2460,2461,2462,2463,2464,2465,2466,2467,2468,2469,2470,2471,2472,2473,2474,2475,2476,2477,2478,2479,2480,2481,2482,2483,2484,2485,2486,2487,2488,2489,2490,2491,2492,2493,2494,2495,2496,2497,2498,2499,2500,2501,2502,2503,2504,2505,2506,2507,2508,2509,2510,2511,2512,2513,2514,2515,2516,2517,2518,2519,2520,2521,2522,2523,2524,2525,2526,2527,2528,2529,2530,2531,2532,2533,2534,2535,2536,2537,2538,2539,2540,2541,2542,2543,2544,2545,2546,2547,2548,2549,2550,2551,2552,2553,2554,2555,2556,2557,2558,2559,3840,3841,3842,3843,3844,3845,3846,3847,3848,3849,3850,3851,3852,3853,3854,3855,3856,3857,3858,3859,3860,3861,3862,3863,3864,3865,3866,3867,3868,3869,3870,3871,3872,3873,3874,3875,3876,3877,3878,3879,3880,3881,3882,3883,3884,3885,3886,3887,3888,3889,3890,3891,3892,3893,3894,3895,3896,3897,3898,3899,3900,3901,3902,3903,3904,3905,3906,3907,3908,3909,3910,3911,3912,3913,3914,3915,3916,3917,3918,3919,3920,3921,3922,3923,3924,3925,3926,3927,3928,3929,3930,3931,3932,3933,3934,3935,3936,3937,3938,3939,3940,3941,3942,3943,3944,3945,3946,3947,3948,3949,3950,3951,3952,3953,3954,3955,3956,3957,3958,3959,3960,3961,3962,3963,3964,3965,3966,3967,4096,4097,4098,4099,4100,4101,4102,4103,4104,4105,4106,4107,4108,4109,4110,4111,4112,4113,4114,4115,4116,4117,4118,4119,4120,4121,4122,4123,4124,4125,4126,4127,4128,4129,4130,4131,4132,4133,4134,4135,4136,4137,4138,4139,4140,4141,4142,4143,4144,4145,4146,4147,4148,4149,4150,4151,4152,4153,4154,4155,4156,4157,4158,4159,4160,4161,4162,4163,4164,4165,4166,4167,4168,4169,4170,4171,4172,4173,4174,4175,4176,4177,4178,4179,4180,4181,4182,4183,4184,4185,4186,4187,4188,4189,4190,4191,4192,4193,4194,4195,4196,4197,4198,4199,4200,4201,4202,4203,4204,4205,4206,4207,4208,4209,4210,4211,4212,4213,4214,4215,4216,4217,4218,4219,4220,4221,4222,4223,5504,5505,5506,5507,5508,5509,5510,5511,5512,5513,5514,5515,5516,5517,5518,5519,5520,5521,5522,5523,5524,5525,5526,5527,5528,5529,5530,5531,5532,5533,5534,5535,5536,5537,5538,5539,5540,5541,5542,5543,5544,5545,5546,5547,5548,5549,5550,5551,5552,5553,5554,5555,5556,5557,5558,5559,5560,5561,5562,5563,5564,5565,5566,5567,5568,5569,5570,5571,5572,5573,5574,5575,5576,5577,5578,5579,5580,5581,5582,5583,5584,5585,5586,5587,5588,5589,5590,5591,5592,5593,5594,5595,5596,5597,5598,5599,5600,5601,5602,5603,5604,5605,5606,5607,5608,5609,5610,5611,5612,5613,5614,5615,5616,5617,5618,5619,5620,5621,5622,5623,5624,5625,5626,5627,5628,5629,5630,5631,4480,4481,4482,4483,4484,4485,4486,4487,4488,4489,4490,4491,4492,4493,4494,4495,4496,4497,4498,4499,4500,4501,4502,4503,4504,4505,4506,4507,4508,4509,4510,4511,4512,4513,4514,4515,4516,4517,4518,4519,4520,4521,4522,4523,4524,4525,4526,4527,4528,4529,4530,4531,4532,4533,4534,4535,4536,4537,4538,4539,4540,4541,4542,4543,4544,4545,4546,4547,4548,4549,4550,4551,4552,4553,4554,4555,4556,4557,4558,4559,4560,4561,4562,4563,4564,4565,4566,4567,4568,4569,4570,4571,4572,4573,4574,4575,4576,4577,4578,4579,4580,4581,4582,4583,4584,4585,4586,4587,4588,4589,4590,4591,4592,4593,4594,4595,4596,4597,4598,4599,4600,4601,4602,4603,4604,4605,4606,4607,5888,5889,5890,5891,5892,5893,5894,5895,5896,5897,5898,5899,5900,5901,5902,5903,5904,5905,5906,5907,5908,5909,5910,5911,5912,5913,5914,5915,5916,5917,5918,5919,5920,5921,5922,5923,5924,5925,5926,5927,5928,5929,5930,5931,5932,5933,5934,5935,5936,5937,5938,5939,5940,5941,5942,5943,5944,5945,5946,5947,5948,5949,5950,5951,5952,5953,5954,5955,5956,5957,5958,5959,5960,5961,5962,5963,5964,5965,5966,5967,5968,5969,5970,5971,5972,5973,5974,5975,5976,5977,5978,5979,5980,5981,5982,5983,5984,5985,5986,5987,5988,5989,5990,5991,5992,5993,5994,5995,5996,5997,5998,5999,6000,6001,6002,6003,6004,6005,6006,6007,6008,6009,6010,6011,6012,6013,6014,6015,6144,6145,6146,6147,6148,6149,6150,6151,6152,6153,6154,6155,6156,6157,6158,6159,6160,6161,6162,6163,6164,6165,6166,6167,6168,6169,6170,6171,6172,6173,6174,6175,6176,6177,6178,6179,6180,6181,6182,6183,6184,6185,6186,6187,6188,6189,6190,6191,6192,6193,6194,6195,6196,6197,6198,6199,6200,6201,6202,6203,6204,6205,6206,6207,6208,6209,6210,6211,6212,6213,6214,6215,6216,6217,6218,6219,6220,6221,6222,6223,6224,6225,6226,6227,6228,6229,6230,6231,6232,6233,6234,6235,6236,6237,6238,6239,6240,6241,6242,6243,6244,6245,6246,6247,6248,6249,6250,6251,6252,6253,6254,6255,6256,6257,6258,6259,6260,6261,6262,6263,6264,6265,6266,6267,6268,6269,6270,6271,7168,7169,7170,7171,7172,7173,7174,7175,7176,7177,7178,7179,7180,7181,7182,7183,7184,7185,7186,7187,7188,7189,7190,7191,7192,7193,7194,7195,7196,7197,7198,7199,7200,7201,7202,7203,7204,7205,7206,7207,7208,7209,7210,7211,7212,7213,7214,7215,7216,7217,7218,7219,7220,7221,72 22,7223,7224,7225,7226,7227,7228,7229,7230,7231,7232,7233,7234,7235,7236,7237,7238,7239,7240,7241,7242,7243,7244,7245,7246, 7247,7248,7249,7250,7251,7252,7253,7254,7255,7256,7257,7258,7259,7260,7261,7262,7263,7264,7265,7266,7267,7268,7269,7270,727 1,7272,7273,7274,7275,7276,7277,7278,7279,7280,7281,7282,7283,7284,7285,7286,7287,7288,7289,7290,7291,7292,7293,7294,7295].

[0137] When N=2048,

[0138] Sequence 1c is:

[0139] Punc_Q=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83 ,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118, 119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,1 50,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,1 81,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,21 2,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243 ,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1536,1537,1538,1539,1540,1541,1542,1543,1544,1545,1546,1547,1548,1549,1550,1551,1552,1553,1554,1555,1556,1557,1558,1559,1560,1561,1562,1563,1564,1565,1566,1567,1568,1569,1570,1571,1572,1573,1574,1575,1576,1577,1578,1579,1580,1581,1582,1583,1584,1585,1586,1587,1588,1589,1590,1591,1592,1593,1594,1595,1596,1597,1598,1599,1600,1601,1602,1603,1604,1605,1606,1607,1608,1609,1610,1611,1612,1613,1614,1615,1616,1617,1618,1619,1620,1621,1622,1623,1624,1625,1626,1627,1628,1629,1630,1631,1632,1633,1634,1635,1636,1637,1638,1639,1640,1641,1642,1643,1644,1645,1646,1647,1648,1649,1650,1651,1652,1653,1654,1655,1656,1657,1658,1659,1660,1661,1662,1663,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,1664,1665,1666,1667,1668,1669,1670,1671,1672,1673,1674,1675,1676,1677,1678,1679,1680,1681,1682,1683,1684,1685,1686,1687,1688,1689,1690,1691,1692,1693,1694,1695,1696,1697,1698,1699,1700,1701,1702,1703,1704,1705,1706,1707,1708,1709,1710,1711,1712,1713,1714,1715,1716,1717,1718,1719,1720,1721,1722,1723,1724,1725,1726,1727,1728,1729,1730,1731,1732,1733,1734,1735,1736,1737,1738,1739,1740,1741,1742,1743,1744,1745,1746,1747,1748,1749,1750,1751,1752,1753,1754,1755,1756,1757,1758,1759,1760,1761,1762,1763,1764,1765,1766,1767,1768,1769,1770,1771,1772,1773,1774,1775,1776,1777,1778,1779,1780,1781,1782,1783,1784,1785,1786,1787,1788,1789,1790,1791,1792,1793,1794,1795,1796,1797,1798,1799,1800,1801,1802,1803,1804,1805,1806,1807,1808,1809,1810,1811,1812,1813,1814,1815,1816,1817,1818,1819,1820,1821,1822,1823,1824,1825,1826,1827,1828,1829,1830,1831,1832,1833,1834,1835,1836,1837,1838,1839,1840,1841,1842,1843,1844,1845,1846,1847,1848,1849,1850,1851,1852,1853,1854,1855,1856,1857,1858,1859,1860,1861,1862,1863,1864,1865,1866,1867,1868,1869,1870,1871,1872,1873,1874,1875,1876,1877,1878,1879,1880,1881,1882,1883,1884,1885,1886,1887,1888,1889,1890,1891,1892,1893,1894,1895,1896,1897,1898,1899,1900,1901,1902,1903,1904,1905,1906,1907,1908,1909,1910,1911,1912,1913,1914,1915,1916,1917,1918,1919,1920,1921,1922,1923,1924,1925,1926,1927,1928,1929,1930,1931,1932,1933,1934,1935,1936,1937,1938,1939,1940,1941,1942,1943,1944,1945,1946,1947,1948,1949,1950,1951,1952,1953,1954,1955,1956,1957,1958,1959,1960,1961,1962,1963,1964,1965,1966,1967,1968,1969,1970,1971,1972,1973,1974,1975,1976,1977,1978,1979,1980,1981,1982,1983,1984,1985,1986,1987,1988,1989,1990,1991,1992,1993,1994,1995,1996,1997,1998,1999,2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,2016,2017,2018,2019,2020,2021,2022,2023,2024,2025,2026,2027,2028,2029,2030,2031,2032,2033,2034,2035,2036,2037,2038,2039,2040,2041,2042,2043,2044,2045,2046,2047].

[0140] Sequence 2c is:

[0141] Punc_Q_half=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895].

[0142] Sequence 3c is;

[0143] Punc_Z=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83 ,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118, 119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,1 50,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,1 81,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,21 2,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243 ,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,2304,2305,2306,2307,2308,2309,2310,2311,2312,2313,2314,2315,2316,2317,2318,2319,2320,2321,2322,2323,2324,2325,2326,2327,2328,2329,2330,2331,2332,2333,2334,2335,2336,2337,2338,2339,2340,2341,2342,2343,2344,2345,2346,2347,2348,2349,2350,2351,2352,2353,2354,2355,2356,2357,2358,2359,2360,2361,2362,2363,2364,2365,2366,2367,2368,2369,2370,2371,2372,2373,2374,2375,2376,2377,2378,2379,2380,2381,2382,2383,2384,2385,2386,2387,2388,2389,2390,2391,2392,2393,2394,2395,2396,2397,2398,2399,2400,2401,2402,2403,2404,2405,2406,2407,2408,2409,2410,2411,2412,2413,2414,2415,2416,2417,2418,2419,2420,2421,2422,2423,2424,2425,2426,2427,2428,2429,2430,2431,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443,1444,1445,1446,1447,1448,1449,1450,1451,1452,1453,1454,1455,1456,1457,1458,1459,1460,1461,1462,1463,1464,1465,1466,1467,1468,1469,1470,1471,1472,1473,1474,1475,1476,1477,1478,1479,1480,1481,1482,1483,1484,1485,1486,1487,1488,1489,1490,1491,1492,1493,1494,1495,1496,1497,1498,1499,1500,1501,1502,1503,1504,1505,1506,1507,1508,1509,1510,1511,1512,1513,1514,1515,1516,1517,1518,1519,1520,1521,1522,1523,1524,1525,1526,1527,1528,1529,1530,1531,1532,1533,1534,1535,2688,2689,2690,2691,2692,2693,2694,2695,2696,2697,2698,2699,2700,2701,2702,2703,2704,2705,2706,2707,2708,2709,2710,2711,2712,2713,2714,2715,2716,2717,2718,2719,2720,2721,2722,2723,2724,2725,2726,2727,2728,2729,2730,2731,2732,2733,2734,2735,2736,2737,2738,2739,2740,2741,2742,2743,2744,2745,2746,2747,2748,2749,2750,2751,2752,2753,2754,2755,2756,2757,2758,2759,2760,2761,2762,2763,2764,2765,2766,2767,2768,2769,2770,2771,2772,2773,2774,2775,2776,2777,2778,2779,2780,2781,2782,2783,2784,2785,2786,2787,2788,2789,2790,2791,2792,2793,2794,2795,2796,2797,2798,2799,2800,2801,2802,2803,2804,2805,2806,2807,2808,2809,2810,2811,2812,2813,2814,2815,1664,1665,1666,1667,1668,1669,1670,1671,1672,1673,1674,1675,1676,1677,1678,1679,1680,1681,1682,1683,1684,1685,1686,1687,1688,1689,1690,1691,1692,1693,1694,1695,1696,1697,1698,1699,1700,1701,1702,1703,1704,1705,1706,1707,1708,1709,1710,1711,1712,1713,1714,1715,1716,1717,1718,1719,1720,1721,1722,1723,1724,1725,1726,1727,1728,1729,1730,1731,1732,1733,1734,1735,1736,1737,1738,1739,1740,1741,1742,1743,1744,1745,1746,1747,1748,1749,1750,1751,1752,1753,1754,1755,1756,1757,1758,1759,1760,1761,1762,1763,1764,1765,1766,1767,1768,1769,1770,1771,1772,1773,1774,1775,1776,1777,1778,1779,1780,1781,1782,1783,1784,1785,1786,1787,1788,1789,1790,1791,2944,2945,2946,2947,2948,2949,2950,2951,2952,2953,2954,2955,2956,2957,2958,2959,2960,2961,2962,2963,2964,2965,2966,2967,2968,2969,2970,2971,2972,2973,2974,2975,2976,2977,2978,2979,2980,2981,2982,2983,2984,2985,2986,2987,2988,2989,2990,2991,2992,2993,2994,2995,2996,2997,2998,2999,3000,3001,3002,3003,3004,3005,3006,3007,3008,3009,3010,3011,3012,3013,3014,3015,3016,3017,3018,3019,3020,3021,3022,3023,3024,3025,3026,3027,3028,3029,3030,3031,3032,3033,3034,3035,3036,3037,3038,3039,3040,3041,3042,3043,3044,3045,3046,3047,3048,3049,3050,3051,3052,3053,3054,3055,3056,3057,3058,3059,3060,3061,3062,3063,3064,3065,3066,3067,3068,3069,3070,3071,3200,3201,3202,3203,3204,3205,3206,3207,3208,3209,3210,3211,3212,3213,3214,3215,3216,3217,3218,3219,3220,3221,3222,3223,3224,3225,3226,3227,3228,3229,3230,3231,3232,3233,3234,3235,3236,3237,3238,3239,3240,3241,3242,3243,3244,3245,3246,3247,3248,3249,3250,3251,3252,3253,3254,3255,3256,3257,3258,3259,3260,3261,3262,3263,3264,3265,3266,3267,3268,3269,3270,3271,3272,3273,3274,3275,3276,3277,3278,3279,3280,3281,3282,3283,3284,3285,3286,3287,3288,3289,3290,3291,3292,3293,3294,3295,3296,3297,3298,3299,3300,3301,3302,3303,3304,3305,3306,3307,3308,3309,3310,3311,3312,3313,3314,3315,3316,3317,3318,3319,3320,3321,3322,3323,3324,3325,3326,3327,4736,4737,4738,4739,4740,4741,4742,4743,4744,4745,4746,4747,4748,4749,4750,4751,4752,4753,4754,4755,4756,4757,4758,4759,4760,4761,4762,4763,4764,4765,4766,4767,4768,4769,4770,4771,4772,4773,4774,4775,4776,4777,4778,4779,4780,4781,4782,4783,4784,4785,4786,4787,4788,4789,4790,4791,4792,4793,4794,4795,4796,4797,4798,4799,4800,4801,4802,4803,4804,4805,4806,4807,4808,4809,4810,4811,4812,4813,4814,4815,4816,4817,4818,4819,4820,4821,4822,4823,4824,4825,482 6,4827,4828,4829,4830,4831,4832,4833,4834,4835,4836,4837,4838,4839,4840,4841,4842,4843,4844,4845,4846,4847,4848,4849,4850,4851,4852,4853,4854,4855,4856,4857,4858,4859,4860,4861,4862,4863].

[0144] When N=1024,

[0145] Sequence 1d is:

[0146] Punc_Q=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83 ,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118, 119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,1 50,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,1 81,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,21 2,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243 ,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023].

[0147] Sequence 2d is:

[0148] Punc_Q_half=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,33 3,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,512,513,514,515,516,517,518,519,520, 521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,58 ,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639].

[0149] Sequence 3d is;

[0150] Punc_Z=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83 ,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118, 119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,1 50,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,1 81,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,21 2,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243 ,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,2304,2305,2306,2307,2308,2309,2310,2311,2312,2313,2314,2315,2316,2317,2318,2319,2320,2321,2322,2323,2324,2325,2326,2327,2328,2329,2330,2331,2332,2333,2334,2335,2336,2337,2338,2339,2340,2341,2342,2343,2344,2345,2346,2347,2348,2349,2350,2351,2352,2353,2354,2355,2356,2357,2358,2359,2360,2361,2362,2363,2364,2365,2366,2367,2368,2369,2370,2371,2372,2373,2374,2375,2376,2377,2378,2379,2380,2381,2382,2383,2384,2385,2386,2387,2388,2389,2390,2391,2392,2393, 2394,2395,2396,2397,2398,2399,2400,2401,2402,2403,2404,2405,2406,2407,2408,2409,2410,2411,2412,2413,2414,2415,2416,2417,2418,2419,2420,2421,2422,2423,2424,2425,2426,2427,2428,2429,2430,2431].

[0151] When N=512,

[0152] Sequence 1e is:

[0153] Punc_Q=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83 ,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118, 119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,1 50,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,1 81,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,21 2,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243 ,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333 34,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,39 3,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452 ,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511].

[0154] Sequence 2e is:

[0155] Punc_Q_half=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37 ,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56, 57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,7 6,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,1 40,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,1 98,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255].

[0156] Sequence 3e is:

[0157] Punc_Z=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, 42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83 ,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118, 119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,1 50,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,1 81,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,21 2,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243 ,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333 34,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,768,769,770,771,772,773,774,775,776,77 7,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836 ,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895].

[0158] Appendix II: Another example of the first sequence:

[0159] Sequence 1:

[0160] Shorten_Q=[0,4096,2048,6144,1024,5120,3072,7168,512,4608,2560,6656,1536,5632,3584,7680,256,4352,2304,6400,1280,5376,3328,7424,768,4864,2816,6912,1792,5888,3840,7936,128,4224,2176,6272,1152,5248,3200,7296,640,4736,2688,6784,1664,5760,3712,7808,384,4480,2432,6528,1408,5504,3456,7552,896,4992,2944,7040,1920,6016,3968,8064,64,4160,2112,6208,1088,5184,3136,7232,576,4672,2624,6720,1600,5696,3648,7744,320,4416,2368,6464,1344,5440,3392,7488,832,4928,2880,6976,1856,5952,3904,8000,192,4288,2240,6336,1216,5312,3264,7360,704,4800,2752,6848,1728,5824,3776,7872,448,4544,2496,6592,1472,5568,3520,7616,960,5056,3008,7104,1984,6080,4032,8128,32,4128,2080,6176,1056,5152,3104,7200,544,4640,2592,6688,1568,5664,3616,7712,288,4384,2336,6432,1312,5408,3360,7456,800,4896,2848,6944,1824,5920,3872,7968,160,4256,2208,6304,1184,5280,3232,7328,672,4768,2720,6816,1696,5792,3744,7840,416,4512,2464,6560,1440,5536,3488,7584,928,5024,2976,7072,1952,6048,4000,8096,96,4192,2144,6240,1120,5216,3168,7264,608,4704,2656,6752,1632,5728,3680,7776,352,4448,2400,6496,1376,5472,3424,7520,864,4960,2912,7008,1888,5984,3936,8032,224,4320,2272,6368,1248,5344,3296,7392,736,4832,2784,6880,1760,5856,3808,7904,480,4576,2528,6624,1504,5600,3552,7648,992,5088,3040,7136,2016,6112,4064,8160,16,4112,2064,6160,1040,5136,3088,7184,528,4624,2576,6672,1552,5648,3600,7696,272,4368,2320,6416,1296,5392,3344,7440,784,4880,2832,6928,1808,5904,3856,7952,144,4240,2192,6288,1168,5264,3216,7312,656,4752,2704,6800,1680,5776,3728,7824,400,4496,2448,6544,1424,5520,3472,7568,912,5008,2960,7056,1936,6032,3984,8080,80,4176,2128,6224,1104,5200,3152,7248,592,4688,2640,6736,1616,5712,3664,7760,336,4432,2384,6480,1360,5456,3408,7504,848,4944,2896,6992,1872,5968,3920,8016,208,4304,2256,6352,1232,5328,3280,7376,720,4816,2768,6864,1744,5840,3792,7888,464,4560,2512,6608,1488,5584,3536,7632,976,5072,3024,7120,2000,6096,4048,8144,48,4144,2096,6192,1072,5168,3120,7216,560,4656,2608,6704,1584,5680,3632,7728,304,4400,2352,6448,1328,5424,3376,7472,816,4912,2864,6960,1840,5936,3888,7984,176,4272,2224,6320,1200,5296,3248,7344,688,4784,2736,6832,1712,5808,3760,7856,432,4528,2480,6576,1456,5552,3504,7600,944,5040,2992,7088,1968,6064,4016,8112,112,4208,2160,6256,1136,5232,3184,7280,624,4720,2672,6768,1648,5744,3696,7792,368,4464,2416,6512,1392,5488,3440,7536,880,4976,2928,7024,1904,6000,3952,8048,240,4336,2288,6384,1264,5360,3312,7408,752,4848,2800,6896,1776,5872,3824,7920,496,4592,2544,6640,1520,5616,3568,7664,1008,5104,3056,7152,2032,6128,4080,8176,8,4104,2056,6152,1032,5128,3080,7176,520,4616,2568,6664,1544,5640,3592,7688,264,4360,2312,6408,1288,5384,3336,7432,776,4872,2824,6920,1800,5896,3848,7944,136,4232,2184,6280,1160,5256,3208,7304,648,4744,2696,6792,1672,5768,3720,7816,392,4488,2440,6536,1416,5512,3464,7560,904,5000,2952,7048,1928,6024,3976,8072,72,4168,2120,6216,1096,5192,3144,7240,584,4680,2632,6728,1608,5704,3656,7752,328,4424,2376,6472,1352,5448,3400,7496,840,4936,2888,6984,1864,5960,3912,8008,200,4296,2248,6344,1224,5320,3272,7368,712,4808,2760,6856,1736,5832,3784,7880,456,4552,2504,6600,1480,5576,3528,7624,968,5064,3016,7112,1992,6088,4040,8136,40,4136,2088,6184,1064,5160,3112,7208,552,4648,2600,6696,1576,5672,3624,7720,296,4392,2344,6440,1320,5416,3368,7464,808,4904,2856,6952,1832,5928,3880,7976,168,4264,2216,6312,1192,5288,3240,7336,680,4776,2728,6824,1704,5800,3752,7848,424,4520,2472,6568,1448,5544,3496,7592,936,5032,2984,7080,1960,6056,4008,8104,104,4200,2152,6248,1128,5224,3176,7272,616,4712,2664,6760,1640,5736,3688,7784,360,4456,2408,6504,1384,5480,3432,7528,872,4968,2920,7016,1896,5992,3944,8040,232,4328,2280,6376,1256,5352,3304,7400,744,4840,2792,6888,1768,5864,3816,7912,488,4584,2536,6632,1512,5608,3560,7656,1000,5096,3048,7144,2024,6120,4072,8168,24,4120,2072,6168,1048,5144,3096,7192,536,4632,2584,6680,1560,5656,3608,7704,280,4376,2328,6424,1304,5400,3352,7448,792,4888,2840,6936,1816,5912,3864,7960,152,4248,2200,6296,1176,5272,3224,7320,664,4760,2712,6808,1688,5784,3736,7832,408,4504,2456,6552,1432,5528,3480,7576,920,5016,2968,7064,1944,6040,3992,8088,88,4184,2136,6232,1112,5208,3160,7256,600,4696,2648,6744,1624,5720,3672,7768,344,4440,2392,6488,1368,5464,3416,7512,856,4952,2904,7000,1880,5976,3928,8024,216,4312,2264,6360,1240,5336,3288,7384,728,4824,2776,6872,1752,5848,3800,7896,472,4568,2520,6616,1496,5592,3544,7640,984,5080,3032,7128,2008,6104,4056,8152,56,4152,2104,6200,1080,5176,3128,7224,568,4664,2616,6712,1592,5688,3640,7736,312,4408,2360,6456,1336,5432,3384,7480,824,4920,2872,6968,1848,5944,3896,7992,184,4280,2232,6328,1208,5304,3256,7352,696,4792,2744,6840,1720,5816,3768,7864,440,4536,2488,6584,1464,5560,3512,7608,952,5048,3000,7096,1976,6072,4024,8120,120,4216,2168,6264,1144,5240,3192,7288,632,4728,2680,6776,1656,5752,3704,7800,376,4472,2424,6520,1400,5496,3448,7544,888,4984,2936,7032,1912,6008,3960,8056,248,4344,2296,6392,1272,5368,3320,7416,760,4856,2808,6904,1784,5880,3832,7928,504,4600,2552,6648,1528,5624,3576,7672,1016,5112,3064,7160,2040,6136,4088,8184,4,4100,2052,6148,1028,5124,3076,7172,516,4612,2564,6660,1540,5636,3588,7684,260,4356,2308,6404,1284,5380,3332,7428,772,4868,2820,6916,1796,5892,3844,7940,132,4228,2180,6276,1156,5252,3204,7300,644,4740,2692,6788,1668,5764,3716,7812,388,4484,2436,6532,1412,5508,3460,7556,900,4996,2948,7044,1924,6020,3972,8068,68,4164,2116,6212,1092,5188,3140,7236,580,4676,2628,6724,1604,5700,3652,7748,324,4420,2372,6468,1348,5444,3396,7492,836,4932,2884,6980,1860,5956,3908,8004,196,4292,2244,6340,1220,5316,3268,7364,708,4804,2756,6852,1732,5828,3780,7876,452,4548,2500,6596,1476,5572,3524,7620,964,5060,3012,7108,1988,6084,4036,8132,36,4132,2084,6180,1060,5156,3108,7204,548,4644,2596,6692,1572,5668,3620,7716,292,4388,2340,6436,1316,5412,3364,7460,804,4900,2852,6948,1828,5924,3876,7972,164,4260,2212,6308,1188,5284,3236,7332,676,4772,2724,6820,1700,5796,3748,7844,420,4516,2468,6564,1444,5540,3492,7588,932,5028,2980,7076,1956,6052,4004,8100,100,4196,2148,6244,1124,5220,3172,7268,612,4708,2660,6756,1636,5732,3684,7780,356,4452,2404,6500,1380,5476,3428,7524,868,4964,2916,7012,1892,5988,3940,8036,228,4324,2276,6372,1252,5348,3300,7396,740,4836,2788,6884,1764,5860,3812,7908,484,4580,2532,6628,1508,5604,3556,7652,996,5092,3044,7140,2020,6116,4068,8164,20,4116,2068,6164,1044,5140,3092,7188,532,4628,2580,6676,1556,5652,3604,7700,276,4372,2324,6420,1300,5396,3348,7444,788,4884,2836,6932,1812,5908,3860,7956,148,4244,2196,6292,1172,5268,3220,7316,660,4756,2708,6804,1684,5780,3732,7828,404,4500,2452,6548,1428,5524,3476,7572,916,5012,2964,7060,1940,6036,3988,8084,84,4180,2132,6228,1108,5204,3156,7252,596,4692,2644,6740,1620,5716,3668,7764,340,4436,2388,6484,1364,5460,3412,7508,852,4948,2900,6996,1876,5972,3924,8020,212,4308,2260,6356,1236,5332,3284,7380,724,4820,2772,6868,1748,5844,3796,7892,468,4564,2516,6612,1492,5588,3540,7636,980,5076,3028,7124,2004,6100,4052,8148,52,4148,2100,6196,1076,5172,3124,7220,564,4660,2612,6708,1588,5684,3636,7732,308,4404,2356,6452,1332,5428,3380,7476,820,4916,2868,6964,1844,5940,3892,7988,180,4276,2228,6324,1204,5300,3252,7348,692,4788,2740,6836,1716,5812,3764,7860,436,4532,2484,6580,1460,5556,3508,7604,948,5044,2996,7092,1972,6068,4020,8116,116,4212,2164,6260,1140,5236,3188,7284,628,4724,2676,6772,1652,5748,3700,7796,372,4468,2420,6516,1396,5492,3444,7540,884,4980,2932,7028,1908,6004,3956,8052,244,4340,2292,6388,1268,5364,3316,7412,756,4852,2804,6900,1780,5876,3828,7924,500,4596,2548,6644,1524,5620,3572,7668,1012,5108,3060,7156,2036,6132,4084,8180,12,4108,2060,6156,1036,5132,3084,7180,524,4620,2572,6668,1548,5644,3596,7692,268,4364,2316,6412,1292,5388,3340,7436,780,4876,2828,6924,1804,5900,3852,7948,140,4236,2188,6284,1164,5260,3212,7308,652,4748,2700,6796,1676,5772,3724,7820,396,4492,2444,6540,1420,5516,3468,7564,908,5004,2956,7052,1932,6028,3980,8076,76,4172,2124,6220,1100,5196,3148,7244,588,4684,2636,6732,1612,5708,3660,7756,332,4428,2380,6476,1356,5452,3404,7500,844,4940,2892,6988,1868,5964,3916,8012,204,4300,2252,6348,1228,5324,3276,7372,716,4812,2764,6860,1740,5836,3788,7884,460,4556,2508,6604,1484,5580,3532,7628,972,5068,3020,7116,1996,6092,4044,8140,44,4140,2092,6188,1068,5164,3116,7212,556,4652,2604,6700,1580,5676,3628,7724,300,4396,2348,6444,1324,5420,3372,7468,812,4908,2860,6956,1836,5932,3884,7980,172,4268,2220,6316,1196,5292,3244,7340,684,4780,2732,6828,1708,5804,3756,7852,428,4524,2476,6572,1452,5548,3500,7596,940,5036,2988,7084,1964,6060,4012,8108,108,4204,2156,6252,1132,5228,3180,7276,620,4716,2668,6764,1644,5740,3692,7788,364,4460,2412,6508,1388,5484,3436,7532,876,4972,2924,7020,1900,5996,3948,8044,236,4332,2284,6380,1260,5356,3308,7404,748,4844,2796,6892,1772,5868,3820,7916,492,4588,2540,6636,1516,5612,3564,7660,1004,5100,3052,7148,2028,6124,4076,8172,28,4124,2076,6172,1052,5148,3100,7196,540,4636,2588,6684,1564,5660,3612,7708,284,4380,2332,6428,1308,5404,3356,7452,796,4892,2844,6940,1820,5916,3868,7964,156,4252,2204,6300,1180,5276,3228,7324,668,4764,2716,6812,1692,5788,3740,7836,412,4508,2460,6556,1436,5532,3484,7580,924,5020,2972,7068,1948,6044,3996,8092,92,4188,2140,6236,1116,5212,3164,7260,604,4700,2652,6748,1628,5724,3676,7772,348,4444,2396,6492,1372,5468,3420,7516,860,4956,2908,7004,1884,5980,3932,8028,220,4316,2268,6364,1244,5340,3292,7388,732,4828,2780,6876,1756,5852,3804,7900,476,4572,2524,6620,1500,5596,3548,7644,988,5084,3036,7132,2012,6108,4060,8156,60,4156,2108,6204,1084,5180,3132,7228,572,4668,2620,6716,1596,5692,3644,7740,316,4412,2364,6460,1340,5436,3388,7484,828,4924,2876,6972,1852,5948,3900,7996,188,4284,2236,6332,1212,5308,3260,7356,700,4796,2748,6844,1724,5820,3772,7868,444,4540,2492,6588,1468,5564,3516,7612,956,5052,3004,7100,1980,6076,4028,8124,124,4220,2172,6268,1148,5244,3196,7292,636,4732,2684,6780,1660,5756,3708,7804,380,4476,2428,6524,1404,5500,3452,7548,892,4988,2940,7036,1916,6012,3964,8060,252,4348,2300,6396,1276,5372,3324,7420,764,4860,2812,6908,1788,5884,3836,7932,508,4604,2556,6652,1532,5628,3580,7676,1020,5116,3068,7164,2044,6140,4092,8188,2,4098,2050,6146,1026,5122,3074,7170,514,4610,2562,6658,1538,5634,3586,7682,258,4354,2306,6402,1282,5378,3330,7426,770,4866,2818,6914,1794,5890,3842,7938,130,4226,2178,6274,1154,5250,3202,7298,642,4738,2690,6786,1666,5762,3714,7810,386,4482,2434,6530,1410,5506,3458,7554,898,4994,2946,7042,1922,6018,3970,8066,66,4162,2114,6210,1090,5186,3138,7234,578,4674,2626,6722,1602,5698,3650,7746,322,4418,2370,6466,1346,5442,3394,7490,834,4930,2882,6978,1858,5954,3906,8002,194,4290,2242,6338,1218,5314,3266,7362,706,4802,2754,6850,1730,5826,3778,7874,450,4546,2498,6594,1474,5570,3522,7618,962,5058,3010,7106,1986,6082,4034,8130,34,4130,2082,6178,1058,5154,3106,7202,546,4642,2594,6690,1570,5666,3618,7714,290,4386,2338,6434,1314,5410,3362,7458,802,4898,2850,6946,1826,5922,3874,7970,162,4258,2210,6306,1186,5282,3234,7330,674,4770,2722,6818,1698,5794,3746,7842,418,4514,2466,6562,1442,5538,3490,7586,930,5026,2978,7074,1954,6050,4002,8098,98,4194,2146,6242,1122,5218,3170,7266,610,4706,2658,6754,1634,5730,3682,7778,354,4450,2402,6498,1378,5474,3426,7522,866,4962,2914,7010,1890,5986,3938,8034,226,4322,2274,6370,1250,5346,3298,7394,738,4834,2786,6882,1762,5858,3810,7906,482,4578,2530,6626,1506,5602,3554,7650,994,5090,3042,7138,2018,6114,4066,8162,18,4114,2066,6162,1042,5138,3090,7186,530,4626,2578,6674,1554,5650,3602,7698,274,4370,2322,6418,1298,5394,3346,7442,786,4882,2834,6930,1810,5906,3858,7954,146,4242,2194,6290,1170,5266,3218,7314,658,4754,2706,6802,1682,5778,3730,7826,402,4498,2450,6546,1426,5522,3474,7570,914,5010,2962,7058,1938,6034,3986,8082,82,4178,2130,6226,1106,5202,3154,7250,594,4690,2642,6738,1618,5714,3666,7762,338,4434,2386,6482,1362,5458,3410,7506,850,4946,2898,6994,1874,5970,3922,8018,210,4306,2258,6354,1234,5330,3282,7378,722,4818,2770,6866,1746,5842,3794,7890,466,4562,2514,6610,1490,5586,3538,7634,978,5074,3026,7122,2002,6098,4050,8146,50,4146,2098,6194,1074,5170,3122,7218,562,4658,2610,6706,1586,5682,3634,7730,306,4402,2354,6450,1330,5426,3378,7474,818,4914,2866,6962,1842,5938,3890,7986,178,4274,2226,6322,1202,5298,3250,7346,690,4786,2738,6834,1714,5810,3762,7858,434,4530,2482,6578,1458,5554,3506,7602,946,5042,2994,7090,1970,6066,4018,8114,114,4210,2162,6258,1138,5234,3186,7282,626,4722,2674,6770,1650,5746,3698,7794,370,4466,2418,6514,1394,5490,3442,7538,882,4978,2930,7026,1906,6002,3954,8050,242,4338,2290,6386,1266,5362,3314,7410,754,4850,2802,6898,1778,5874,3826,7922,498,4594,2546,6642,1522,5618,3570,7666,1010,5106,3058,7154,2034,6130,4082,8178,10,4106,2058,6154,1034,5130,3082,7178,522,4618,2570,6666,1546,5642,3594,7690,266,4362,2314,6410,1290,5386,3338,7434,778,4874,2826,6922,1802,5898,3850,7946,138,4234,2186,6282,1162,5258,3210,7306,650,4746,2698,6794,1674,5770,3722,7818,394,4490,2442,6538,1418,5514,3466,7562,906,5002,2954,7050,1930,6026,3978,8074,74,4170,2122,6218,1098,5194,3146,7242,586,4682,2634,6730,1610,5706,3658,7754,330,4426,2378,6474,1354,5450,3402,7498,842,4938,2890,6986,1866,5962,3914,8010,202,4298,2250,6346,1226,5322,3274,7370,714,4810,2762,6858,1738,5834,3786,7882,458,4554,2506,6602,1482,5578,3530,7626,970,5066,3018,7114,1994,6090,4042,8138,42,4138,2090,6186,1066,5162,3114,7210,554,4650,2602,6698,1578,5674,3626,7722,298,4394,2346,6442,1322,5418,3370,7466,810,4906,2858,6954,1834,5930,3882,7978,170,4266,2218,6314,1194,5290,3242,7338,682,4778,2730,6826,1706,5802,3754,7850,426,4522,2474,6570,1450,5546,3498,7594,938,5034,2986,7082,1962,6058,4010,8106,106,4202,2154,6250,1130,5226,3178,7274,618,4714,2666,6762,1642,5738,3690,7786,362,4458,2410,6506,1386,5482,3434,7530,874,4970,2922,7018,1898,5994,3946,8042,234,4330,2282,6378,1258,5354,3306,7402,746,4842,2794,6890,1770,5866,3818,7914,490,4586,2538,6634,1514,5610,3562,7658,1002,5098,3050,7146,2026,6122,4074,8170,26,4122,2074,6170,1050,5146,3098,7194,538,4634,2586,6682,1562,5658,3610,7706,282,4378,2330,6426,1306,5402,3354,7450,794,4890,2842,6938,1818,5914,3866,7962,154,4250,2202,6298,1178,5274,3226,7322,666,4762,2714,6810,1690,5786,3738,7834,410,4506,2458,6554,1434,5530,3482,7578,922,5018,2970,7066,1946,6042,3994,8090,90,4186,2138,6234,1114,5210,3162,7258,602,4698,2650,6746,1626,5722,3674,7770,346,4442,2394,6490,1370,5466,3418,7514,858,4954,2906,7002,1882,5978,3930,8026,218,4314,2266,6362,1242,5338,3290,7386,730,4826,2778,6874,1754,5850,3802,7898,474,4570,2522,6618,1498,5594,3546,7642,986,5082,3034,7130,2010,6106,4058,8154,58,4154,2106,6202,1082,5178,3130,7226,570,4666,2618,6714,1594,5690,3642,7738,314,4410,2362,6458,1338,5434,3386,7482,826,4922,2874,6970,1850,5946,3898,7994,186,4282,2234,6330,1210,5306,3258,7354,698,4794,2746,6842,1722,5818,3770,7866,442,4538,2490,6586,1466,5562,3514,7610,954,5050,3002,7098,1978,6074,4026,8122,122,4218,2170,6266,1146,5242,3194,7290,634,4730,2682,6778,1658,5754,3706,7802,378,4474,2426,6522,1402,5498,3450,7546,890,4986,2938,7034,1914,6010,3962,8058,250,4346,2298,6394,1274,5370,3322,7418,762,4858,2810,6906,1786,5882,3834,7930,506,4602,2554,6650,1530,5626,3578,7674,1018,5114,3066,7162,2042,6138,4090,8186,6,4102,2054,6150,1030,5126,3078,7174,518,4614,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79,3331,7427,771,4867,2819,6915,1795,5891,3843,7939,131,4227,2179,6275,1155,5251,3203,7299,643,4739,2691,6787,1667,5763,3715,7811,387,4483,2435,6531,1411,5507,3459,7555,899,4995,2947,7043,1923,6019,3971,8067,67,4163,2115,6211,1091,5187,3139,7235,579,4675,2627,6723,1603,5699,3651,7747,323,4419,2371,6467,1347,5443,3395,7491,835,4931,2883,6979,1859,5955,3907,8003,195,4291,2243,6339,1219,5315,3267,7363,707,4803,2755,6851,1731,5827,3779,7875,451,4547,2499,6595,1475,5571,3523,7619,963,5059,3011,7107,1987,6083,4035,8131,35,4131,2083,6179,1059,5155,3107,7203,547,4643,2595,6691,1571,5667,3619,7715,291,4387,2339,6435,1315,5411,3363,7459,803,4899,2851,6947,1827,5923,3875,7971,163,4259,2211,6307,1187,5283,3235,7331,675,4771,2723,6819,1699,5795,3747,7843,419,4515,2467,6563,1443,5539,3491,7587,931,5027,2979,7075,1955,6051,4003,8099,99,4195,2147,6243,1123,5219,3171,7267,611,4707,2659,6755,1635,5731,3683,7779,355,4451,2403,6499,1379,5475,3427,7523,867,4963,2915,7011,1891,5987,3939,8035,227,4323,2275,6371,1251,5347,3299,7395,739,4835,2787,6883,1763,5859,3811,7907,483,4579,2531,6627,1507,5603,3555,7651,995,5091,3043,7139,2019,6115,4067,8163,19,4115,2067,6163,1043,5139,3091,7187,531,4627,2579,6675,1555,5651,3603,7699,275,4371,2323,6419,1299,5395,3347,7443,787,4883,2835,6931,1811,5907,3859,7955,147,4243,2195,6291,1171,5267,3219,7315,659,4755,2707,6803,1683,5779,3731,7827,403,4499,2451,6547,1427,5523,3475,7571,915,5011,2963,7059,1939,6035,3987,8083,83,4179,2131,6227,1107,5203,3155,7251,595,4691,2643,6739,1619,5715,3667,7763,339,4435,2387,6483,1363,5459,3411,7507,851,4947,2899,6995,1875,5971,3923,8019,211,4307,2259,6355,1235,5331,3283,7379,723,4819,2771,6867,1747,5843,3795,7891,467,4563,2515,6611,1491,5587,3539,7635,979,5075,3027,7123,2003,6099,4051,8147,51,4147,2099,6195,1075,5171,3123,7219,563,4659,2611,6707,1587,5683,3635,7731,307,4403,2355,6451,1331,5427,3379,7475,819,4915,2867,6963,1843,5939,3891,7987,179,4275,2227,6323,1203,5299,3251,7347,691,4787,2739,6835,1715,5811,3763,7859,435,4531,2483,6579,1459,5555,3507,7603,947,5043,2995,7091,1971,6067,4019,8115,115,4211,2163,6259,1139,5235,3187,7283,627,4723,2675,6771,1651,5747,3699,7795,371,4467,2419,6515,1395,5491,3443,7539,883,4979,2931,7027,1907,6003,3955,8051,243,4339,2291,6387,1267,5363,3315,7411,755,4851,2803,6899,1779,5875,3827,7923,499,4595,2547,6643,1523,5619,3571,7667,1011,5107,3059,7155,2035,6131,4083,8179,11,4107,2059,6155,1035,5131,3083,7179,523,4619,2571,6667,1547,5643,3595,7691,267,4363,2315,6411,1291,5387,3339,7435,779,4875,2827,6923,1803,5899,3851,7947,139,4235,2187,6283,1163,5259,3211,7307,651,4747,2699,6795,1675,5771,3723,7819,395,4491,2443,6539,1419,5515,3467,7563,907,5003,2955,7051,1931,6027,3979,8075,75,4171,2123,6219,1099,5195,3147,7243,587,4683,2635,6731,1611,5707,3659,7755,331,4427,2379,6475,1355,5451,3403,7499,843,4939,2891,6987,1867,5963,3915,8011,203,4299,2251,6347,1227,5323,3275,7371,715,4811,2763,6859,1739,5835,3787,7883,459,4555,2507,6603,1483,5579,3531,7627,971,5067,3019,7115,1995,6091,4043,8139,43,4139,2091,6187,1067,5163,3115,7211,555,4651,2603,6699,1579,5675,3627,7723,299,4395,2347,6443,1323,5419,3371,7467,811,4907,2859,6955,1835,5931,3883,7979,171,4267,2219,6315,1195,5291,3243,7339,683,4779,2731,6827,1707,5803,3755,7851,427,4523,2475,6571,1451,5547,3499,7595,939,5035,2987,7083,1963,6059,4011,8107,107,4203,2155,6251,1131,5227,3179,7275,619,4715,2667,6763,1643,5739,3691,7787,363,4459,2411,6507,1387,5483,3435,7531,875,4971,2923,7019,1899,5995,3947,8043,235,4331,2283,6379,1259,5355,3307,7403,747,4843,2795,6891,1771,5867,3819,7915,491,4587,2539,6635,1515,5611,3563,7659,1003,5099,3051,7147,2027,6123,4075,8171,27,4123,2075,6171,1051,5147,3099,7195,539,4635,2587,6683,1563,5659,3611,7707,283,4379,2331,6427,1307,5403,3355,7451,795,4891,2843,6939,1819,5915,3867,7963,155,4251,2203,6299,1179,5275,3227,7323,667,4763,2715,6811,1691,5787,3739,7835,411,4507,2459,6555,1435,5531,3483,7579,923,5019,2971,7067,1947,6043,3995,8091,91,4187,2139,6235,1115,5211,3163,7259,603,4699,2651,6747,1627,5723,3675,7771,347,4443,2395,6491,1371,5467,3419,7515,859,4955,2907,7003,1883,5979,3931,8027,219,4315,2267,6363,1243,5339,3291,7387,731,4827,2779,6875,1755,5851,3803,7899,475,4571,2523,6619,1499,5595,3547,7643,987,5083,3035,7131,2011,6107,4059,8155,59,4155,2107,6203,1083,5179,3131,7227,571,4667,2619,6715,1595,5691,3643,7739,315,4411,2363,6459,1339,5435,3387,7483,827,4923,2875,6971,1851,5947,3899,7995,187,4283,2235,6331,1211,5307,3259,7355,699,4795,2747,6843,1723,5819,3771,7867,443,4539,2491,6587,1467,5563,3515,7611,955,5051,3003,7099,1979,6075,4027,8123,123,4219,2171,6267,1147,5243,3195,7291,635,4731,2683,6779,1659,5755,3707,7803,379,4475,2427,6523,1403,5499,3451,7547,891,4987,2939,7035,1915,6011,3963,8059,251,4347,2299,6395,1275,5371,3323,7419,763,4859,2811,6907,1787,5883,3835,7931,507,4603,2555,6651,1531,5627,3579,7675,1019,5115,3067,7163,2043,6139,4091,8187,7,4103,2055,6151,1031,5127,3079,7175,519,4615,2567,6663,1543,5639,3591,7687,263,4359,2311,6407,1287,5383,3335,7431,775,4871,2823,6919,1799,5895,3847,7943,135,4231,2183,6279,1159,5255,3207,7303,647,4743,2695,6791,1671,5767,3719,7815,391,4487,2439,6535,1415,5511,3463,7559,903,4999,2951,7047,1927,6023,3975,8071,71,4167,2119,6215,1095,5191,3143,7239,583,4679,2631,6727,1607,5703,3655,7751,327,4423,2375,6471,1351,5447,3399,7495,839,4935,2887,6983,1863,5959,3911,8007,199,4295,2247,6343,1223,5319,3271,7367,711,4807,2759,6855,1735,5831,3783,7879,455,4551,2503,6599,1479,5575,3527,7623,967,5063,3015,7111,1991,6087,4039,8135,39,4135,2087,6183,1063,5159,3111,7207,551,4647,2599,6695,1575,5671,3623,7719,295,4391,2343,6439,1319,5415,3367,7463,807,4903,2855,6951,1831,5927,3879,7975,167,4263,2215,6311,1191,5287,3239,7335,679,4775,2727,6823,1703,5799,3751,7847,423,4519,2471,6567,1447,5543,3495,7591,935,5031,2983,7079,1959,6055,4007,8103,103,4199,2151,6247,1127,5223,3175,7271,615,4711,2663,6759,1639,5735,3687,7783,359,4455,2407,6503,1383,5479,3431,7527,871,4967,2919,7015,1895,5991,3943,8039,231,4327,2279,6375,1255,5351,3303,7399,743,4839,2791,6887,1767,5863,3815,7911,487,4583,2535,6631,1511,5607,3559,7655,999,5095,3047,7143,2023,6119,4071,8167,23,4119,2071,6167,1047,5143,3095,7191,535,4631,2583,6679,1559,5655,3607,7703,279,4375,2327,6423,1303,5399,3351,7447,791,4887,2839,6935,1815,5911,3863,7959,151,4247,2199,6295,1175,5271,3223,7319,663,4759,2711,6807,1687,5783,3735,7831,407,4503,2455,6551,1431,5527,3479,7575,919,5015,2967,7063,1943,6039,3991,8087,87,4183,2135,6231,1111,5207,3159,7255,599,4695,2647,6743,1623,5719,3671,7767,343,4439,2391,6487,1367,5463,3415,7511,855,4951,2903,6999,1879,5975,3927,8023,215,4311,2263,6359,1239,5335,3287,7383,727,4823,2775,6871,1751,5847,3799,7895,471,4567,2519,6615,1495,5591,3543,7639,983,5079,3031,7127,2007,6103,4055,8151,55,4151,2103,6199,1079,5175,3127,7223,567,4663,2615,6711,1591,5687,3639,7735,311,4407,2359,6455,1335,5431,3383,7479,823,4919,2871,6967,1847,5943,3895,7991,183,4279,2231,6327,1207,5303,3255,7351,695,4791,2743,6839,1719,5815,3767,7863,439,4535,2487,6583,1463,5559,3511,7607,951,5047,2999,7095,1975,6071,4023,8119,119,4215,2167,6263,1143,5239,3191,7287,631,4727,2679,6775,1655,5751,3703,7799,375,4471,2423,6519,1399,5495,3447,7543,887,4983,2935,7031,1911,6007,3959,8055,247,4343,2295,6391,1271,5367,3319,7415,759,4855,2807,6903,1783,5879,3831,7927,503,4599,2551,6647,1527,5623,3575,7671,1015,5111,3063,7159,2039,6135,4087,8183,15,4111,2063,6159,1039,5135,3087,7183,527,4623,2575,6671,1551,5647,3599,7695,271,4367,2319,6415,1295,5391,3343,7439,783,4879,2831,6927,1807,5903,3855,7951,143,4239,2191,6287,1167,5263,3215,7311,655,4751,2703,6799,1679,5775,3727,7823,399,4495,2447,6543,1423,5519,3471,7567,911,5007,2959,7055,1935,6031,3983,8079,79,4175,2127,6223,1103,5199,3151,7247,591,4687,2639,6735,1615,5711,3663,7759,335,4431,2383,6479,1359,5455,3407,7503,847,4943,2895,6991,1871,5967,3919,8015,207,4303,2255,6351,1231,5327,3279,7375,719,4815,2767,6863,1743,5839,3791,7887,463,4559,2511,6607,1487,5583,3535,7631,975,5071,3023,7119,1999,6095,4047,8143,47,4143,2095,6191,1071,5167,3119,7215,559,4655,2607,6703,1583,5679,3631,7727,303,4399,2351,6447,1327,5423,3375,7471,815,4911,2863,6959,1839,5935,3887,7983,175,4271,2223,6319,1199,5295,3247,7343,687,4783,2735,6831,1711,5807,3759,7855,431,4527,2479,6575,1455,5551,3503,7599,943,5039,2991,7087,1967,6063,4015,8111,111,4207,2159,6255,1135,5231,3183,7279,623,4719,2671,6767,1647,5743,3695,7791,367,4463,2415,6511,1391,5487,3439,7535,879,4975,2927,7023,1903,5999,3951,8047,239,4335,2287,6383,1263,5359,3311,7407,751,4847,2799,6895,1775,5871,3823,7919,495,4591,2543,6639,1519,5615,3567,7663,1007,5103,3055,7151,2031,6127,4079,8175,31,4127,2079,6175,1055,5151,3103,7199,543,4639,2591,6687,1567,5663,3615,7711,287,4383,2335,6431,1311,5407,3359,7455,799,4895,2847,6943,1823,5919,3871,7967,159,4255,2207,6303,1183,5279,3231,7327,671,4767,2719,6815,1695,5791,3743,7839,415,4511,2463,6559,1439,5535,3487,7583,927,5023,2975,7071,1951,6047,3999,8095,95,4191,2143,6239,1119,5215,3167,7263,607,4703,2655,6751,1631,5727,3679,7775,351,4447,2399,6495,1375,5471,3423,7519,863,4959,2911,7007,1887,5983,3935,8031,223,4319,2271,6367,1247,5343,3295,7391,735,4831,2783,6879,1759,5855,3807,7903,479,4575,2527,6623,1503,5599, 3551,7647,991,5087,3039,7135,2015,6111,4063,8159,63,4159,2111,6207,1087,5183,3135,7231,575,4671,2623,6719,1599,5695,3647,7743,319,4415,2367,6463,1343,5439,3391,7487,831,4927,2879,6975,1855,5951,3903,7999,191,4287,2239,6335,12 15,5311,3263,7359,703,4799,2751,6847,1727,5823,3775,7871,447,4543,2495,6591,1471,5567,3519,7615,959,5055,3007,7103,1983,6079,4031,8127,127,4223,2175,6271,1151,5247,3199,7295,639,4735,2687,6783,1663,5759,3711,7807,383,4479,24 31,6527,1407,5503,3455,7551,895,4991,2943,7039,1919,6015,3967,8063,255,4351,2303,6399,1279,5375,3327,7423,767,4863,2815,6911,1791,5887,3839,7935,511,4607,2559,6655,1535,5631,3583,7679,1023,5119,3071,7167,2047,6143,4095,8191].

[0161] Sequence 2:

[0162] Shorten_Q_half=[1,4097,2049,6145,1025,5121,3073,7169,513,4609,2561,6657,1537,5633,3585,7681,257,4353,2305,6401,1281,5377,3329,7425,769,4865,2817,6913,1793,5889,3841,7937,129,4225,2177,6273,1153,5249,3201,7297,641,4737,2689,6785,1665,5761,3713,7809,385,4481,2433,6529,1409,5505,3457,7553,897,4993,2945,7041,1921,6017,3969,8065,65,4161,2113,6209,1089,5185,3137,7233,577,4673,2625,6721,1601,5697,3649,7745,321,4417,2369,6465,1345,5441,3393,7489,833,4929,2881,6977,1857,5953,3905,8001,193,4289,2241,6337,1217,5313,3265,7361,705,4801,2753,6849,1729,5825,3777,7873,449,4545,2497,6593,1473,5569,3521,7617,961,5057,3009,7105,1985,6081,4033,8129,33,4129,2081,6177,1057,5153,3105,7201,545,4641,2593,6689,1569,5665,3617,7713,289,4385,2337,6433,1313,5409,3361,7457,801,4897,2849,6945,1825,5921,3873,7969,161,4257,2209,6305,1185,5281,3233,7329,673,4769,2721,6817,1697,5793,3745,7841,417,4513,2465,6561,1441,5537,3489,7585,929,5025,2977,7073,1953,6049,4001,8097,97,4193,2145,6241,1121,5217,3169,7265,609,4705,2657,6753,1633,5729,3681,7777,353,4449,2401,6497,1377,5473,3425,7521,865,4961,2913,7009,1889,5985,3937,8033,225,4321,2273,6369,1249,5345,3297,7393,737,4833,2785,6881,1761,5857,3809,7905,481,4577,2529,6625,1505,5601,3553,7649,993,5089,3041,7137,2017,6113,4065,8161,17,4113,2065,6161,1041,5137,3089,7185,529,4625,2577,6673,1553,5649,3601,7697,273,4369,2321,6417,1297,5393,3345,7441,785,4881,2833,6929,1809,5905,3857,7953,145,4241,2193,6289,1169,5265,3217,7313,657,4753,2705,6801,1681,5777,3729,7825,401,4497,2449,6545,1425,5521,3473,7569,913,5009,2961,7057,1937,6033,3985,8081,81,4177,2129,6225,1105,5201,3153,7249,593,4689,2641,6737,1617,5713,3665,7761,337,4433,2385,6481,1361,5457,3409,7505,849,4945,2897,6993,1873,5969,3921,8017,209,4305,2257,6353,1233,5329,3281,7377,721,4817,2769,6865,1745,5841,3793,7889,465,4561,2513,6609,1489,5585,3537,7633,977,5073,3025,7121,2001,6097,4049,8145,49,4145,2097,6193,1073,5169,3121,7217,561,4657,2609,6705,1585,5681,3633,7729,305,4401,2353,6449,1329,5425,3377,7473,817,4913,2865,6961,1841,5937,3889,7985,177,4273,2225,6321,1201,5297,3249,7345,689,4785,2737,6833,1713,5809,3761,7857,433,4529,2481,6577,1457,5553,3505,7601,945,5041,2993,7089,1969,6065,4017,8113,113,4209,2161,6257,1137,5233,3185,7281,625,4721,2673,6769,1649,5745,3697,7793,369,4465,2417,6513,1393,5489,3441,7537,881,4977,2929,7025,1905,6001,3953,8049,241,4337,2289,6385,1265,5361,3313,7409,753,4849,2801,6897,1777,5873,3825,7921,497,4593,2545,6641,1521,5617,3569,7665,1009,5105,3057,7153,2033,6129,4081,8177,9,4105,2057,6153,1033,5129,3081,7177,521,4617,2569,6665,1545,5641,3593,7689,265,4361,2313,6409,1289,5385,3337,7433,777,4873,2825,6921,1801,5897,3849,7945,137,4233,2185,6281,1161,5257,3209,7305,649,4745,2697,6793,1673,5769,3721,7817,393,4489,2441,6537,1417,5513,3465,7561,905,5001,2953,7049,1929,6025,3977,8073,73,4169,2121,6217,1097,5193,3145,7241,585,4681,2633,6729,1609,5705,3657,7753,329,4425,2377,6473,1353,5449,3401,7497,841,4937,2889,6985,1865,5961,3913,8009,201,4297,2249,6345,1225,5321,3273,7369,713,4809,2761,6857,1737,5833,3785,7881,457,4553,2505,6601,1481,5577,3529,7625,969,5065,3017,7113,1993,6089,4041,8137,41,4137,2089,6185,1065,5161,3113,7209,553,4649,2601,6697,1577,5673,3625,7721,297,4393,2345,6441,1321,5417,3369,7465,809,4905,2857,6953,1833,5929,3881,7977,169,4265,2217,6313,1193,5289,3241,7337,681,4777,2729,6825,1705,5801,3753,7849,425,4521,2473,6569,1449,5545,3497,7593,937,5033,2985,7081,1961,6057,4009,8105,105,4201,2153,6249,1129,5225,3177,7273,617,4713,2665,6761,1641,5737,3689,7785,361,4457,2409,6505,1385,5481,3433,7529,873,4969,2921,7017,1897,5993,3945,8041,233,4329,2281,6377,1257,5353,3305,7401,745,4841,2793,6889,1769,5865,3817,7913,489,4585,2537,6633,1513,5609,3561,7657,1001,5097,3049,7145,2025,6121,4073,8169,25,4121,2073,6169,1049,5145,3097,7193,537,4633,2585,6681,1561,5657,3609,7705,281,4377,2329,6425,1305,5401,3353,7449,793,4889,2841,6937,1817,5913,3865,7961,153,4249,2201,6297,1177,5273,3225,7321,665,4761,2713,6809,1689,5785,3737,7833,409,4505,2457,6553,1433,5529,3481,7577,921,5017,2969,7065,1945,6041,3993,8089,89,4185,2137,6233,1113,5209,3161,7257,601,4697,2649,6745,1625,5721,3673,7769,345,4441,2393,6489,1369,5465,3417,7513,857,4953,2905,7001,1881,5977,3929,8025,217,4313,2265,6361,1241,5337,3289,7385,729,4825,2777,6873,1753,5849,3801,7897,473,4569,2521,6617,1497,5593,3545,7641,985,5081,3033,7129,2009,6105,4057,8153,57,4153,2105,6201,1081,5177,3129,7225,569,4665,2617,6713,1593,5689,3641,7737,313,4409,2361,6457,1337,5433,3385,7481,825,4921,2873,6969,1849,5945,3897,7993,185,4281,2233,6329,1209,5305,3257,7353,697,4793,2745,6841,1721,5817,3769,7865,441,4537,2489,6585,1465,5561,3513,7609,953,5049,3001,7097,1977,6073,4025,8121,121,4217,2169,6265,1145,5241,3193,7289,633,4729,2681,6777,1657,5753,3705,7801,377,4473,2425,6521,1401,5497,3449,7545,889,4985,2937,7033,1913,6009,3961,8057,249,4345,2297,6393,1273,5369,3321,7417,761,4857,2809,6905,1785,5881,3833,7929,505,4601,2553,6649,1529,5625,3577,7673,1017,5113,3065,7161,2041,6137,4089,8185,5,4101,2053,6149,1029,5125,3077,7173,517,4613,2565,6661,1541,5637,3589,7685,261,4357,2309,6405,1285,5381,3333,7429,773,4869,2821,6917,1797,5893,3845,7941,133,4229,2181,6277,1157,5253,3205,7301,645,4741,2693,6789,1669,5765,3717,7813,389,4485,2437,6533,1413,5509,3461,7557,901,4997,2949,7045,1925,6021,3973,8069,69,4165,2117,6213,1093,5189,3141,7237,581,4677,2629,6725,1605,5701,3653,7749,325,4421,2373,6469,1349,5445,3397,7493,837,4933,2885,6981,1861,5957,3909,8005,197,4293,2245,6341,1221,5317,3269,7365,709,4805,2757,6853,1733,5829,3781,7877,453,4549,2501,6597,1477,5573,3525,7621,965,5061,3013,7109,1989,6085,4037,8133,37,4133,2085,6181,1061,5157,3109,7205,549,4645,2597,6693,1573,5669,3621,7717,293,4389,2341,6437,1317,5413,3365,7461,805,4901,2853,6949,1829,5925,3877,7973,165,4261,2213,6309,1189,5285,3237,7333,677,4773,2725,6821,1701,5797,3749,7845,421,4517,2469,6565,1445,5541,3493,7589,933,5029,2981,7077,1957,6053,4005,8101,101,4197,2149,6245,1125,5221,3173,7269,613,4709,2661,6757,1637,5733,3685,7781,357,4453,2405,6501,1381,5477,3429,7525,869,4965,2917,7013,1893,5989,3941,8037,229,4325,2277,6373,1253,5349,3301,7397,741,4837,2789,6885,1765,5861,3813,7909,485,4581,2533,6629,1509,5605,3557,7653,997,5093,3045,7141,2021,6117,4069,8165,21,4117,2069,6165,1045,5141,3093,7189,533,4629,2581,6677,1557,5653,3605,7701,277,4373,2325,6421,1301,5397,3349,7445,789,4885,2837,6933,1813,5909,3861,7957,149,4245,2197,6293,1173,5269,3221,7317,661,4757,2709,6805,1685,5781,3733,7829,405,4501,2453,6549,1429,5525,3477,7573,917,5013,2965,7061,1941,6037,3989,8085,85,4181,2133,6229,1109,5205,3157,7253,597,4693,2645,6741,1621,5717,3669,7765,341,4437,2389,6485,1365,5461,3413,7509,853,4949,2901,6997,1877,5973,3925,8021,213,4309,2261,6357,1237,5333,3285,7381,725,4821,2773,6869,1749,5845,3797,7893,469,4565,2517,6613,1493,5589,3541,7637,981,5077,3029,7125,2005,6101,4053,8149,53,4149,2101,6197,1077,5173,3125,7221,565,4661,2613,6709,1589,5685,3637,7733,309,4405,2357,6453,1333,5429,3381,7477,821,4917,2869,6965,1845,5941,3893,7989,181,4277,2229,6325,1205,5301,3253,7349,693,4789,2741,6837,1717,5813,3765,7861,437,4533,2485,6581,1461,5557,3509,7605,949,5045,2997,7093,1973,6069,4021,8117,117,4213,2165,6261,1141,5237,3189,7285,629,4725,2677,6773,1653,5749,3701,7797,373,4469,2421,6517,1397,5493,3445,7541,885,4981,2933,7029,1909,6005,3957,8053,245,4341,2293,6389,1269,5365,3317,7413,757,4853,2805,6901,1781,5877,3829,7925,501,4597,2549,6645,1525,5621,3573,7669,1013,5109,3061,7157,2037,6133,4085,8181,13,4109,2061,6157,1037,5133,3085,7181,525,4621,2573,6669,1549,5645,3597,7693,269,4365,2317,6413,1293,5389,3341,7437,781,4877,2829,6925,1805,5901,3853,7949,141,4237,2189,6285,1165,5261,3213,7309,653,4749,2701,6797,1677,5773,3725,7821,397,4493,2445,6541,1421,5517,3469,7565,909,5005,2957,7053,1933,6029,3981,8077,77,4173,2125,6221,1101,5197,3149,7245,589,4685,2637,6733,1613,5709,3661,7757,333,4429,2381,6477,1357,5453,3405,7501,845,4941,2893,6989,1869,5965,3917,8013,205,4301,2253,6349,1229,5325,3277,7373,717,4813,2765,6861,1741,5837,3789,7885,461,4557,2509,6605,1485,5581,3533,7629,973,5069,3021,7117,1997,6093,4045,8141,45,4141,2093,6189,1069,5165,3117,7213,557,4653,2605,6701,1581,5677,3629,7725,301,4397,2349,6445,1325,5421,3373,7469,813,4909,2861,6957,1837,5933,3885,7981,173,4269,2221,6317,1197,5293,3245,7341,685,4781,2733,6829,1709,5805,3757,7853,429,4525,2477,6573,1453,5549,3501,7597,941,5037,2989,7085,1965,6061,4013,8109,109,4205,2157,6253,1133,5229,3181,7277,621,4717,2669,6765,1645,5741,3693,7789,365,4461,2413,6509,1389,5485,3437,7533,877,4973,2925,7021,1901,5997,3949,8045,237,4333,2285,6381,1261,5357,3309,7405,749,4845,2797,6893,1773,5869,3821,7917,493,4589,2541,6637,1517,5613,3565,7661,1005,5101,3053,7149,2029,6125,4077,8173,29,4125,2077,6173,1053,5149,3101,7197,541,4637,2589,6685,1565,5661,3613,7709,285,4381,2333,6429,1309,5405,3357,7453,797,4893,2845,6941,1821,5917,3869,7965,157,4253,2205,6301,1181,5277,3229,7325,669,4765,2717,6813,1693,5789,3741,7837,413,4509,2461,6557,1437,5533,3485,7581,925,5021,2973,7069,1949,6045,3997,8093,93,4189,2141,6237,1117,5213,3165,7261,605,4701,2653,6749,1629,5725,3677,7773,349,4445,2397,6493,1373,5469,3421,7517,861,4957,2909,7005,1885,5981,3933,8029,221,4317,2269,6365,1245,5341,3293,7389,733,4829,2781,6877,1757,5853,3805,7901,477,4573,2525,6621,1501,5597,3549,7645,989,5085,3037,7133,2013,6109,4061,8157,61,4157,2109,6205,1085,5181,3133,7229,573,4669,2621,6717,1597,5693,3645,7741,317,4413,2365,6461,1341,5437,3389,7485,829,4925,2877,6973,1853,5949,3901,7997,189,4285,2237,6333,1213,5309,3261,7357,701,4797,2749,6845,1725,5821,3773,7869,445,4541,2493,6589,1469,5565,3517,7613,957,5053,3005,7101,1981,6077,4029,8125,125,4221,2173,6269,1149,5245,3197,7293,637,4733,2685,6781,1661,5757,3709,7805,381,4477,2429,6525,1405,5501,3453,7549,893,4989,2941,7037,1917,6013,3965,8061,253,4349,2301,6397,1277,5373,3325,7421,765,4861,2813,6909,1789,5885,3837,7933,509,4605,2557,6653,1533,5629,3581,7677,1021,5117,3069,7165,2045,6141,4093,8189,3,4099,2051,6147,1027,5123,3075,7171,515,4611,2563,6659,1539,5635,3587,7683,259,4355,2307,6403,1283,5379,3331,7427,771,4867,2819,6915,1795,5891,3843,7939,131,4227,2179,6275,1155,5251,3203,7299,643,4739,2691,6787,1667,5763,3715,7811,387,4483,2435,6531,1411,5507,3459,7555,899,4995,2947,7043,1923,6019,3971,8067,67,4163,2115,6211,1091,5187,3139,7235,579,4675,2627,6723,1603,5699,3651,7747,323,4419,2371,6467,1347,5443,3395,7491,835,4931,2883,6979,1859,5955,3907,8003,195,4291,2243,6339,1219,5315,3267,7363,707,4803,2755,6851,1731,5827,3779,7875,451,4547,2499,6595,1475,5571,3523,7619,963,5059,3011,7107,1987,6083,4035,8131,35,4131,2083,6179,1059,5155,3107,7203,547,4643,2595,6691,1571,5667,3619,7715,291,4387,2339,6435,1315,5411,3363,7459,803,4899,2851,6947,1827,5923,3875,7971,163,4259,2211,6307,1187,5283,3235,7331,675,4771,2723,6819,1699,5795,3747,7843,419,4515,2467,6563,1443,5539,3491,7587,931,5027,2979,7075,1955,6051,4003,8099,99,4195,2147,6243,1123,5219,3171,7267,611,4707,2659,6755,1635,5731,3683,7779,355,4451,2403,6499,1379,5475,3427,7523,867,4963,2915,7011,1891,5987,3939,8035,227,4323,2275,6371,1251,5347,3299,7395,739,4835,2787,6883,1763,5859,3811,7907,483,4579,2531,6627,1507,5603,3555,7651,995,5091,3043,7139,2019,6115,4067,8163,19,4115,2067,6163,1043,5139,3091,7187,531,4627,2579,6675,1555,5651,3603,7699,275,4371,2323,6419,1299,5395,3347,7443,787,4883,2835,6931,1811,5907,3859,7955,147,4243,2195,6291,1171,5267,3219,7315,659,4755,2707,6803,1683,5779,3731,7827,403,4499,2451,6547,1427,5523,3475,7571,915,5011,2963,7059,1939,6035,3987,8083,83,4179,2131,6227,1107,5203,3155,7251,595,4691,2643,6739,1619,5715,3667,7763,339,4435,2387,6483,1363,5459,3411,7507,851,4947,2899,6995,1875,5971,3923,8019,211,4307,2259,6355,1235,5331,3283,7379,723,4819,2771,6867,1747,5843,3795,7891,467,4563,2515,6611,1491,5587,3539,7635,979,5075,3027,7123,2003,6099,4051,8147,51,4147,2099,6195,1075,5171,3123,7219,563,4659,2611,6707,1587,5683,3635,7731,307,4403,2355,6451,1331,5427,3379,7475,819,4915,2867,6963,1843,5939,3891,7987,179,4275,2227,6323,1203,5299,3251,7347,691,4787,2739,6835,1715,5811,3763,7859,435,4531,2483,6579,1459,5555,3507,7603,947,5043,2995,7091,1971,6067,4019,8115,115,4211,2163,6259,1139,5235,3187,7283,627,4723,2675,6771,1651,5747,3699,7795,371,4467,2419,6515,1395,5491,3443,7539,883,4979,2931,7027,1907,6003,3955,8051,243,4339,2291,6387,1267,5363,3315,7411,755,4851,2803,6899,1779,5875,3827,7923,499,4595,2547,6643,1523,5619,3571,7667,1011,5107,3059,7155,2035,6131,4083,8179,11,4107,2059,6155,1035,5131,3083,7179,523,4619,2571,6667,1547,5643,3595,7691,267,4363,2315,6411,1291,5387,3339,7435,779,4875,2827,6923,1803,5899,3851,7947,139,4235,2187,6283,1163,5259,3211,7307,651,4747,2699,6795,1675,5771,3723,7819,395,4491,2443,6539,1419,5515,3467,7563,907,5003,2955,7051,1931,6027,3979,8075,75,4171,2123,6219,1099,5195,3147,7243,587,4683,2635,6731,1611,5707,3659,7755,331,4427,2379,6475,1355,5451,3403,7499,843,4939,2891,6987,1867,5963,3915,8011,203,4299,2251,6347,1227,5323,3275,7371,715,4811,2763,6859,1739,5835,3787,7883,459,4555,2507,6603,1483,5579,3531,7627,971,5067,3019,7115,1995,6091,4043,8139,43,4139,2091,6187,1067,5163,3115,7211,555,4651,2603,6699,1579,5675,3627,7723,299,4395,2347,6443,1323,5419,3371,7467,811,4907,2859,6955,1835,5931,3883,7979,171,4267,2219,6315,1195,5291,3243,7339,683,4779,2731,6827,1707,5803,3755,7851,427,4523,2475,6571,1451,5547,3499,7595,939,5035,2987,7083,1963,6059,4011,8107,107,4203,2155,6251,1131,5227,3179,7275,619,4715,2667,6763,1643,5739,3691,7787,363,4459,2411,6507,1387,5483,3435,7531,875,4971,2923,7019,1899,5995,3947,8043,235,4331,2283,6379,1259,5355,3307,7403,747,4843,2795,6891,1771,5867,3819,7915,491,4587,2539,6635,1515,5611,3563,7659,1003,5099,3051,7147,2027,6123,4075,8171,27,4123,2075,6171,1051,5147,3099,7195,539,4635,2587,6683,1563,5659,3611,7707,283,4379,2331,6427,1307,5403,3355,7451,795,4891,2843,6939,1819,5915,3867,7963,155,4251,2203,6299,1179,5275,3227,7323,667,4763,2715,6811,1691,5787,3739,7835,411,4507,2459,6555,1435,5531,3483,7579,923,5019,2971,7067,1947,6043,3995,8091,91,4187,2139,6235,1115,5211,3163,7259,603,4699,2651,6747,1627,5723,3675,7771,347,4443,2395,6491,1371,5467,3419,7515,859,4955,2907,7003,1883,5979,3931,8027,219,4315,2267,6363,1243,5339,3291,7387,731,4827,2779,6875,1755,5851,3803,7899,475,4571,2523,6619,1499,5595,3547,7643,987,5083,3035,7131,2011,6107,4059,8155,59,4155,2107,6203,1083,5179,3131,7227,571,4667,2619,6715,1595,5691,3643,7739,315,4411,2363,6459,1339,5435,3387,7483,827,4923,2875,6971,1851,5947,3899,7995,187,4283,2235,6331,1211,5307,3259,7355,699,4795,2747,6843,1723,5819,3771,7867,443,4539,2491,6587,1467,5563,3515,7611,955,5051,3003,7099,1979,6075,4027,8123,123,4219,2171,6267,1147,5243,3195,7291,635,4731,2683,6779,1659,5755,3707,7803,379,4475,2427,6523,1403,5499,3451,7547,891,4987,2939,7035,1915,6011,3963,8059,251,4347,2299,6395,1275,5371,3323,7419,763,4859,2811,6907,1787,5883,3835,7931,507,4603,2555,6651,1531,5627,3579,7675,1019,5115,3067,7163,2043,6139,4091,8187,7,4103,2055,6151,1031,5127,3079,7175,519,4615,2567,6663,1543,5639,3591,7687,263,4359,2311,6407,1287,5383,3335,7431,775,4871,2823,6919,1799,5895,3847,7943,135,4231,2183,6279,1159,5255,3207,7303,647,4743,2695,6791,1671,5767,3719,7815,391,4487,2439,6535,1415,5511,3463,7559,903,4999,2951,7047,1927,6023,3975,8071,71,4167,2119,6215,1095,5191,3143,7239,583,4679,2631,6727,1607,5703,3655,7751,327,4423,2375,6471,1351,5447,3399,7495,839,4935,2887,6983,1863,5959,3911,8007,199,4295,2247,6343,1223,5319,3271,7367,711,4807,2759,6855,1735,5831,3783,7879,455,4551,2503,6599,1479,5575,3527,7623,967,5063,3015,7111,1991,6087,4039,8135,39,4135,2087,6183,1063,5159,3111,7207,551,4647,2599,6695,1575,5671,3623,7719,295,4391,2343,6439,1319,5415,3367,7463,807,4903,2855,6951,1831,5927,3879,7975,167,4263,2215,6311,1191,5287,3239,7335,679,4775,2727,6823,1703,5799,3751,7847,423,4519,2471,6567,1447,5543,3495,7591,935,5031,2983,7079,1959,6055,4007,8103,103,4199,2151,6247,1127,5223,3175,7271,615,4711,2663,6759,1639,5735,3687,7783,359,4455,2407,6503,1383,5479,3431,7527,871,4967,2919,7015,1895,5991,3943,8039,231,4327,2279,6375,1255,5351,3303,7399,743,4839,2791,6887,1767,5863,3815,7911,487,4583,2535,6631,1511,5607,3559,7655,999,5095,3047,7143,2023,6119,4071,8167,23,4119,2071,6167,1047,5143,3095,7191,535,4631,2583,6679,1559,5655,3607,7703,279,4375,2327,6423,1303,5399,3351,7447,791,4887,2839,6935,1815,5911,3863,7959,151,4247,2199,6295,1175,5271,3223,7319,663,4759,2711,6807,1687,5783,3735,7831,407,4503,2455,6551,1431,5527,3479,7575,919,5015,2967,7063,1943,6039,3991,8087,87,4183,2135,6231,1111,5207,3159,7255,599,4695,2647,6743,1623,5719,3671,7767,343,4439,2391,6487,1367,5463,3415,7511,855,4951,2903,6999,1879,5975,3927,8023,215,4311,2263,6359,1239,5335,3287,7383,727,4823,2775,6871,1751,5847,3799,7895,471,4567,2519,6615,1495,5591,3543,7639,983,5079,3031,7127,2007,6103,4055,8151,55,4151,2103,6199,1079,5175,3127,7223,567,4663,2615,6711,1591,5687,3639,7735,311,4407,2359,6455,1335,5431,3383,7479,823,4919,2871,6967,1847,5943,3895,7991,183,4279,2231,6327,1207,5303,3255,7351,695,4791,2743,6839,1719,5815,3767,7863,439,4535,2487,6583,1463,5559,3511,7607,951,5047,2999,7095,1975,6071,4023,8119,119,4215,2167,6263,1143,5239,3191,7287,631,4727,2679,6775,1655,5751,3703,7799,375,4471,2423,6519,1399,5495,3447,7543,887,4983,2935,7031,1911,6007,3959,8055,247,4343,2295,6391,1271,5367,3319,7415,759,4855,2807,6903,1783,5879,3831,7927,503,4599,2551,6647,1527,5623,3575,7671,1015,5111,3063,7159,2039,6135,4087,8183,15,4111,2063,6159,1039,5135,3087,7183,527,4623,2575,6671,1551,5647,3599,7695,271,4367,2319,6415,1295,5391,3343,7439,783,4879,2831,6927,1807,5903,3855,7951,143,4239,2191,6287,1167,5263,3215,7311,655,4751,2703,6799,1679,5775,3727,7823,399,4495,2447,6543,1423,5519,3471,7567,911,5007,2959,7055,1935,6031,3983,8079,79,4175,2127,6223,1103,5199,3151,7247,591,4687,2639,6735,1615,5711,3663,7759,335,4431,2383,6479,1359,5455,3407,7503,847,4943,2895,6991,1871,5967,3919,8015,207,4303,2255,6351,1231,5327,3279,7375,719,4815,2767,6863,1743,5839,3791,7887,463,4559,2511,6607,1487,5583,3535,7631,975,5071,3023,7119,1999,6095,4047,8143,47,4143,2095,6191,1071,5167,3119,7215,559,4655,2607,6703,1583,5679,3631,7727,303,4399,2351,6447,1327,5423,3375,7471,815,4911,2863,6959,1839,5935,3887,7983,175,4271,2223,6319,1199,5295,3247,7343,687,4783,2735,6831,1711,5807,3759,7855,431,4527,2479,6575,1455,5551,3503,7599,943,5039,2991,7087,1967,6063,4015,8111,111,4207,2159,6255,1135,5231,3183,7279,623,4719,2671,6767,1647,5743,3695,7791,367,4463,2415,6511,1391,5487,3439,7535,879,4975,2927,7023,1903,5999,3951,8047,239,4335,2287,6383,1263,5359,3311,7407,751,4847,2799,6895,1775,5871,3823,7919,495,4591,2543,6639,1519,5615,3567,7663,1007,5103,3055,7151,2031,6127,4079,8175,31,4127,2079,6175,1055,5151,3103,7199,543,4639,2591,6687,1567,5663,3615,7711,287,4383,2335,6431,1311,5407,3359,7455,799,4895,2847,6943,1823,5919,3871,7967,159,4255,2207,6303,1183,5279,3231,7327,671,4767,2719,6815,1695,5791,3743,7839,415,4511,2463,6559,1439,5535,3487,7583,927,5023,2975,7071,1951,6047,3999,8095,95,4191,2143,6239,1119,5215,3167,7263,607,4703,2655,6751,1631,5727,3679,7775,351,4447,2399,6495,1375,5471,3423,7519,863,4959,2911,7007,1887,5983,3935,8031,223,4319,2271,6367,1247,5343,3295,7391,735,4831,2783,6879,17 59,5855,3807,7903,479,4575,2527,6623,1503,5599,3551,7647,991,5087,3039,7135,2015,6111,4063,8159,63,4159,2111,6207,1087,5183,3135,7231,575,4671,2623,6719,1599,5695,3647,7743,319,4415,2367,6463,1343,5439,3391,7487,831,4927,2879,6975,1855,5951 ,3903,7999,191,4287,2239,6335,1215,5311,3263,7359,703,4799,2751,6847,1727,5823,3775,7871,447,4543,2495,6591,1471,5567,3519,7615,959,5055,3007,7103,1983,6079,4031,8127,127,4223,2175,6271,1151,5247,3199,7295,639,4735,2687,6783,1663,5759,3711, 7807,383,4479,2431,6527,1407,5503,3455,7551,895,4991,2943,7039,1919,6015,3967,8063,255,4351,2303,6399,1279,5375,3327,7423,767,4863,2815,6911,1791,5887,3839,7935,511,4607,2559,6655,1535,5631,3583,7679,1023,5119,3071,7167,2047,6143,4095,8191].

[0163] Sequence 3:

[0164] Shorten_Z=[0,4096,2048,6144,1024,5120,3072,7168,512,4608,2560,6656,1536,5632,3584,7680,256,4352,2304,6400,1280,5376,3328,7424,768,4864,2816,6912,1792,5888,3840,7936,128,4224,2176,6272,1152,5248,3200,7296,640,4736,2688,6784,1664,5760,3712,7808,384,4480,2432,6528,1408,5504,3456,7552,896,4992,2944,7040,1920,6016,3968,8064,64,4160,2112,6208,1088,5184,3136,7232,576,4672,2624,6720,1600,5696,3648,7744,320,4416,2368,6464,1344,5440,3392,7488,832,4928,2880,6976,1856,5952,3904,8000,192,4288,2240,6336,1216,5312,3264,7360,704,4800,2752,6848,1728,5824,3776,7872,448,4544,2496,6592,1472,5568,3520,7616,960,5056,3008,7104,1984,6080,4032,8128,32,4128,2080,6176,1056,5152,3104,7200,544,4640,2592,6688,1568,5664,3616,7712,288,4384,2336,6432,1312,5408,3360,7456,800,4896,2848,6944,1824,5920,3872,7968,160,4256,2208,6304,1184,5280,3232,7328,672,4768,2720,6816,1696,5792,3744,7840,416,4512,2464,6560,1440,5536,3488,7584,928,5024,2976,7072,1952,6048,4000,8096,96,4192,21...

Claims

1. A rate matching method for polar code encoding, characterized in that: include: Get K bits to be encoded, target code length M, mother code length N, N= , M, K, n and N are positive integers; Performing polar coding on the K to-be-coded bits according to the mother code length N to obtain a coded bit sequence; Performing rate matching on the encoded bit sequence according to a rate matching sequence to obtain a rate-matched sequence; Output the rate-matched sequence.

2. A method for rate matching for polar code decoding, characterized in that: include: Obtaining an information sequence to be decoded comprising M bits, including K bits to be decoded; Determine the mother code length N, mother code length N, N= , M, K, n and N are positive integers; performing rate matching and decoding on the information sequence to be decoded according to the mother code length N and the rate matching sequence to obtain a decoding result; The decoding result is output.

3. The method according to claim 1 or 2, wherein: The rate matching sequence is a subsequence of a first sequence, where the first sequence is sequence 1, sequence 2, or sequence 3 in Appendix 1, wherein sequence 1 includes N2 polarization subchannel numbers, sequence 2 includes N2 / 2 polarization subchannel numbers, and sequence 3 includes N2 values ​​of different sizes. The polarization subchannel numbers start from 0, and N2 is greater than or equal to N.

4. The method according to any one of claims 1 to 3, wherein When N=8192, the rate matching sequence is sequence 1a, sequence 2a, or sequence 3a in the specification; or When N=4096, the rate matching sequence is sequence 1b, 2b, or 3b in the specification; or When N=2048, the rate matching sequence is sequence 1c, 2c, or 3c in the specification; or When N=1024, the rate matching sequence is sequence 1d, 2d, or 3d in the specification; or When N=512, the rate matching sequence is the sequence 1e, 2e, or 3e in the specification; wherein the sequence 1a, the sequence 1b, the sequence 1c, the sequence 1d, and the sequence 1e are subsequences of the sequence 1; Sequence 2a, sequence 2b, sequence 2c, sequence 2d, and sequence 2e are subsequences of sequence 2; The sequence 3a, the sequence 3b, the sequence 3c, the sequence 3d, and the sequence 3e are subsequences of the sequence 3.

5. The method according to claim 3 or 4, wherein: The rate matching method is a puncture operation. If the first sequence is sequence 1, the puncturing operation is: taking bits carried by subchannels corresponding to the last M polarized subchannel numbers in the rate matching sequence according to the target code length M, or deleting bits carried by subchannels corresponding to the first NM polarized subchannel numbers in the rate matching sequence according to the target code length M; or If the first sequence is sequence 2, the puncturing operation is: deleting bits carried by subchannels corresponding to the first NM polarized subchannel numbers in the rate matching sequence according to the target code length M; or If the first sequence is sequence 3, the puncturing operation is: deleting bits carried by the polarization subchannels containing the smallest NM values ​​in the rate matching sequence according to the target code length M, or taking bits carried by the polarization subchannels containing the largest M values ​​in the rate matching sequence according to the target code length M.

6. The method according to any one of claims 1 to 5, wherein The K bits include cyclic redundancy check bits.

7. The method according to any one of claims 1 to 6, wherein The K bits include parity check bits.

8. The method according to any one of claims 1, 3 to 7, wherein: The polarization coding of the K to-be-coded bits according to the mother code length N includes mapping the K to-be-coded bits to K polarization subchannels with higher reliability among the MP non-punctured polarization subchannels, such that the reliability of the polarization subchannel containing the K to-be-coded bits is higher than the reliability of the remaining MPK polarization subchannels, where P is the number of pre-frozen polarization subchannels, and P is greater than or equal to 0.

9. The method according to any one of claims 1, 3 to 8, wherein Before performing rate matching on the encoded bit sequence according to the rate matching sequence to obtain a rate-matched sequence, the encoded bit sequence is interleaved.

10. A polar code rate matching device, characterized in that: include: Acquisition unit, K bits to be encoded, target code length M, where K is a positive integer; The encoding unit is configured to determine a mother code length N, and to perform polarization encoding on the K bits to be encoded according to the mother code length N to obtain an encoded bit sequence, where N=2 n , n is a positive integer, K≤N; as well as The rate matching unit is configured to perform rate matching on the encoded bit sequence according to a rate matching sequence to obtain a rate-matched bit sequence of length M.

11. A Polar code decoding rate matching device, characterized in that: include: An acquisition unit 1001 is configured to acquire an information sequence to be decoded; The processing unit 1002 is configured to determine a mother code length N, and perform rate matching and decoding and demodulation operations on the information sequence to be decoded according to the mother code length N and the rate matching sequence, thereby obtaining and outputting a decoding result.

12. The device according to claim 10 or 11, characterized in that The rate matching sequence is a subsequence of a first sequence, where the first sequence is sequence 1, sequence 2, or sequence 3 in Appendix 1, wherein sequence 1 includes N2 polarization subchannel numbers, sequence 2 includes N2 / 2 polarization subchannel numbers, and sequence 3 includes N2 values ​​of different sizes. The polarization subchannel numbers start from 0, and N2 is greater than or equal to N.

13. The device according to any one of claims 10 to 12, characterized in that When N=8192, the rate matching sequence is sequence 1a, sequence 2a, or sequence 3a in the specification; or When N=4096, the rate matching sequence is sequence 1b, 2b, or 3b in the specification; or When N=2048, the rate matching sequence is sequence 1c, 2c, or 3c in the specification; or When N=1024, the rate matching sequence is sequence 1d, 2d, or 3d in the specification; or When N=512, the rate matching sequence is the sequence 1e, 2e, or 3e in the specification; wherein the sequence 1a, the sequence 1b, the sequence 1c, the sequence 1d, and the sequence 1e are subsequences of the sequence 1; Sequence 2a, sequence 2b, sequence 2c, sequence 2d, and sequence 2e are subsequences of sequence 2; The sequence 3a, the sequence 3b, the sequence 3c, the sequence 3d, and the sequence 3e are subsequences of the sequence 3.

14. The device according to claim 12 or 13, characterized in that The rate matching method is a puncture operation. If the first sequence is sequence 1, the puncturing operation is: taking bits carried by subchannels corresponding to the last M polarized subchannel numbers in the rate matching sequence according to the target code length M, or deleting bits carried by subchannels corresponding to the first NM polarized subchannel numbers in the rate matching sequence according to the target code length M; or If the first sequence is sequence 2, the puncturing operation is: deleting bits carried by subchannels corresponding to the first NM polarized subchannel numbers in the rate matching sequence according to the target code length M; or If the first sequence is sequence 3, the puncturing operation is: deleting bits carried by the polarized subchannels containing the NM smallest values ​​in the rate matching sequence according to the target code length M, or extracting bits carried by the polarized subchannels containing the M largest values ​​in the rate matching sequence according to the target code length M.

15. The device according to any one of claims 10 to 14, characterized in that The K bits include cyclic redundancy check bits.

16. The device according to any one of claims 10 to 15, characterized in that The K bits include parity check bits.

17. The device according to any one of claims 10, 12 to 16, characterized in that: The polarization coding of the K to-be-coded bits according to the mother code length N includes mapping the K to-be-coded bits to K polarization subchannels with higher reliability among the MP non-punctured polarization subchannels, such that the reliability of the polarization subchannel containing the K to-be-coded bits is higher than the reliability of the remaining MPK polarization subchannels, where P is the number of pre-frozen polarization subchannels, and P is greater than or equal to 0.

18. The device according to any one of claims 10, 12 to 17, characterized in that: Before performing rate matching on the encoded bit sequence according to the rate matching sequence to obtain a rate-matched bit sequence with a length of M, the encoded bit sequence is interleaved.

19. A polar code rate matching device, characterized in that: include: A processor, configured to execute the method according to any one of claims 1, 3 to 8.

20. The device according to claim 19, wherein Also includes: A memory for storing program instructions used by the processor to execute the method according to any one of claims 1 and 3 to 8.

21. A Polar code rate matching device, characterized in that: include: An input interface circuit for obtaining K bits to be encoded; A logic circuit, configured to obtain a rate-matched sequence based on the K bits to be coded according to the method according to any one of claims 1 and 3 to 8; The output interface circuit is used to output the rate-matched sequence.

22. A polar code decoding rate matching device, characterized in that: include: A processor, configured to execute the method according to any one of claims 2 to 7.

23. The device according to claim 19, wherein Also includes: A memory for storing program instructions used by the processor to execute the method according to any one of claims 2 to 7.

24. A Polar code decoding rate matching device, characterized in that: include: An input interface circuit for obtaining an information sequence to be decoded; a logic circuit configured to perform a rate matching operation and a decoding and demodulation operation on the information sequence to be decoded according to the method according to any one of claims 2 to 7, thereby obtaining a decoding result; The output interface circuit is used to output the decoding result.

25. A computer-readable medium, characterized in that Computer program instructions are stored, and when the computer program instructions are executed by a computer, the method according to any one of claims 1 to 9 is performed.

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