Quasi-linear time complexity coding and unique decoding algorithm of any-order Reed-Muller code and application

Abstract

The invention discloses a quasi-linear time complexity coding and unique decoding algorithm for any-order Reed-Muller codes, which comprises the following steps of: defining a q-element RMq (r, m) RM code, defining information and converting the information into a unitary information polynomial; coding is carried out by using an RS code coder, so that MPE acceleration of an m-element polynomial on # imgabs0 # is realized, and the coding complexity reaches a quasi-linear level; and inputting the received code word into a unique decoder of the RS code, calculating to obtain a correct message polynomial, solving an m-element information polynomial through a mapping relation, and taking a coefficient as a unique decoding result. According to the method, on the basis of the thought of transformation between a multivariate polynomial ring on # imgabs1 # and a univariate polynomial ring on an extended finite field # imgabs2 #, a quasi-linear time algorithm for unique decoding of any r-order RM codes is realized, coding and decoding of the RM codes are accelerated, and the applicability of the coding scheme is enhanced.

Description

Technical Field

[0001] The present invention belongs to the field of communication technology, and particularly relates to a quasi-linear time complexity encoding and unique decoding algorithm and application of arbitrary order Reed-Muller (hereinafter referred to as RM) code. Background Art

[0002] Error correction codes are coding schemes used to reliably transmit information over noisy communication channels. When information is transmitted over various lossy channels, it is often necessary to consider coding that is resistant to noise. is a finite field The linear error correction code on the message space The linear encoding is The code words in . The code rate is the ratio of the code dimension to the code length, that is, K / N. The minimum distance is The minimum Hamming distance between any two different codewords in Among them, the code rate and the time complexity of encoding and decoding are important indicators to measure its performance.

[0003] When it is assumed that the number of errors e occurring in the communication channel is less than half of the minimum coding distance, the original codeword can be uniquely recovered, which is unique decoding.

[0004] RM code was introduced by Muller in 1954, and later Reed proposed a unique decoding algorithm with an error correction radius of half the minimum distance. The encoding of RM code is actually multi-variable multi-point assignment (hereinafter referred to as MPE). It is a daunting challenge to design an approximately linear time algorithm for the input size of MPE. In current research, the complexity of the quasi-linear unique decoding algorithm of almost all known RM codes still has room for optimization.

[0005] Through the above analysis, the problems and defects of the existing technology are: the encoding and decoding of RM codes has always been an important research topic in coding theory and theoretical computer science. Although researchers have made great efforts before, the quasi-linear time algorithm of arbitrary-order RM codes still has a large room for optimization, and research results in this direction are still scarce. Summary of the invention

[0006] In view of the problem of implementing the quasi-linear time algorithm for encoding and decoding RM codes in the background technology, the present invention proposes a solution that can realize encoding and decoding of any r-order RM code with quasi-linear time complexity, as follows:

[0007] A quasi-linear time complexity encoding and unique decoding algorithm for arbitrary order Reed-Muller codes, including:

[0008] Define q-ary RM q (r,m)RM code, defines information and converts it into a univariate information polynomial;

[0009] Will The m-variable MPE problem on the extended domain is simplified to The univariate MPE problem on , using the RS code encoder to encode, so as to achieve The MPE acceleration of the upper m-dimensional polynomial makes the coding complexity reach the quasi-linear level;

[0010] The received codeword is input into the unique decoder of the RS code, the correct message polynomial is calculated, and the m-ary information polynomial is obtained through the mapping relationship, and the coefficients are the unique decoding results.

[0011] Further, the RS code encoder is used for encoding, thereby realizing The MPE acceleration of the m-dimensional polynomial makes the encoding complexity reach the quasi-linear level. The specific steps are as follows:

[0012] Take the q-element RM in Definition 4 q The code length of (r,m)RM code is N = q m , the dimension is m i Based on Definition 7, the definition information is

[0013] Mapping information into information polynomials in Based on Definition 7;

[0014] Encoder calculation using RS code The returned c is the final encoded codeword.

[0015] Furthermore, the definition 4: defines the q-ary RM code according to the MPE of definition 3:

[0016]

[0017] Its code length is N = q m , the dimension is The minimum code distance is d = q m-1 (qr), is the set defined in Definition 2.

[0018] Furthermore, the definition 3: multivariable multipoint assignment (hereinafter referred to as MPE) is defined as follows: given N assignment point sets Take the polynomial F in Definition 1 and the set in Definition 2 And calculate (F(a1),…,F(a N )):

[0019]

[0020] Furthermore, the definition 1: define a finite field polynomials of m variables on the Define the total degree of F as:

[0021] Furthermore, the definition 2: definition set are all polynomials with a total degree less than or equal to r subspace, the dimension is

[0022] Furthermore, the above Definition 7: Let m i is the value in definition 6 above. The period, m i Divides m and has subdomain set up yes A normal basis of , for every 1≤i≤s and 1≤j≤m in Definition 6 i , define the polynomial:

[0023]

[0024] Furthermore, the definition 6: Take m in definition 4, including q module m The cyclotomic coset of -1 is defined as:

[0025] U λ = {q i λmod(q m -1)|1≤i≤m}

[0026] if Different, and but A subset of {λ1,λ2,…,λ s} is called q module m The complete set represented by the cyclotomic coset of -1.

[0027] Further, The RM code on the The RS code on the , thus making the decoding complexity of any r-order RM code reach the quasi-linear level, the specific steps are as follows:

[0028] The received codeword Input to the RS code decoder to calculate the correct message polynomial

[0029] Calculated in The m-variable polynomial F = τ(f(x));

[0030] The coefficient of the final output F is the unique decoding result.

[0031] The present invention also provides an application of a quasi-linear time complexity encoding and unique decoding algorithm of an arbitrary order Reed-Muller code, characterized in that the application is applicable to technical fields including military communications and complex channel transmission.

[0032] In combination with the above technical solutions and the technical problems solved, the advantages and positive effects of the technical solutions to be protected by the present invention are as follows:

[0033] First, the present invention is based on Multivariate Polynomial Rings and Extended Finite Fields on The key advantages of this approach include:

[0034] 1. Accelerate RM code encoding and decoding: The present invention is based on Multivariate Polynomial Rings and Extended Finite Fields on The idea of ​​transformation between univariate polynomial rings on The m-variable MPE problem on the extended domain is simplified to The univariate MPE problem on The RM code on the The RS code on the RM code can be uniquely decoded, so that the encoding and decoding complexity of the RM code reaches a quasi-linear level, speeding up the encoding and decoding time.

[0035] 2. A quasi-linear time algorithm for uniquely decoding any r-order RM code is realized: The present invention is based on Multivariate Polynomial Rings and Extended Finite Fields on The idea of ​​transformation between univariate polynomial rings on The RM code on the The RS code on is uniquely decoded, thus realizing a quasi-linear time algorithm for unique decoding of any r-order RM code.

[0036] 3. Wide applicability: This method is applicable to industrial communication fields that require high encoding and decoding time complexity, including but not limited to secure communication, sensitive information transmission, and steganography.

[0037] The present invention not only embodies a significant technical advancement, but also provides an effective improvement and supplement to the existing RM code encoding and decoding. By implementing this method, the RM code encoding and decoding time can be accelerated to meet the growing demand for secure communication and coping with complex channels.

[0038] Second, the present invention is based on Multivariate Polynomial Rings and Extended Finite Fields on The idea of ​​transforming between univariate polynomial rings on a binary system has accelerated the encoding and decoding of RM codes and achieved significant technological progress.

[0039] Accelerating RM code encoding and decoding: This method reduces the encoding and decoding time complexity to a quasi-linear level, thereby speeding up the encoding and decoding time.

[0040] A quasi-linear time algorithm for unique decoding of arbitrary-order RM codes is implemented: This algorithm implements a quasi-linear time algorithm for unique decoding of arbitrary-order RM codes.

[0041] Strong adaptability: The present invention is applicable to RM codes of various parameters and various application scenarios.

[0042] Technological innovation: The encoding and decoding method of the present invention is an effective supplement and improvement to the existing RM code encoding and decoding technology, demonstrating technological innovation in the field of communications.

[0043] Third, the expected benefits and commercial value of the technical solution of the present invention after transformation are:

[0044] The information is encoded with RM code through this scheme and used in scenarios in the industry that require high encoding and decoding speed. This technology can be used in fields such as secure communication, sensitive information transmission and steganography.

[0045] The technical solution of the present invention fills the technical gap in the industry at home and abroad: the quasi-linear time algorithm of arbitrary-order RM codes still has a large room for optimization. To fill this gap, the present invention is based on Multivariate Polynomial Rings and Extended Finite Fields on The idea of ​​transformation between univariate polynomial rings on The m-variable MPE problem on the extended domain is simplified to The univariate MPE problem on The RM code on the The RS code on can be uniquely decoded, thus making the encoding and decoding complexity of any r-order RM code reach a quasi-linear level.

[0046] The technical solution of the present invention solves a technical problem that people have been eager to solve but have never been able to solve successfully: the method realizes a quasi-linear time algorithm for decoding RM codes of arbitrary orders and optimizes existing theoretical achievements. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 The flowchart is a quasi-linear time complexity encoding and unique decoding algorithm of arbitrary-order RM codes of the present invention. DETAILED DESCRIPTION

[0048] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0049] In view of the problems existing in the prior art, the present invention provides a quasi-linear time algorithm solution for implementing encoding and decoding of arbitrary-order RM codes. The present invention is described in detail below.

[0050] The scheme is based on Multivariate Polynomial Rings and Extended Finite Fields on Transformations between univariate polynomial rings on , 1. The m-variable MPE problem on the extended domain is simplified to The univariate MPE problem on ; 2. The RM code on the The Reed-Solomon (hereinafter referred to as RS) code on the .

[0051] The proof of this algorithm requires some definitions and theorems as follows:

[0052] Definition 1: Define a finite field polynomials of m variables on the Define the total degree of F as:

[0053] Definition 2: Define a set are all polynomials with a total degree less than or equal to r subspace, the dimension is

[0054] Definition 3: Multivariable multi-point assignment (hereinafter referred to as MPE) is defined as follows: Given N assignment point sets Take the polynomial F in Definition 1 and the set in Definition 2 And calculate (F(a1),…,F(a N )):

[0055]

[0056] Definition 4: Based on the MPE definition of Definition 3, we define a q-ary RM code:

[0057]

[0058] Its code length is N = q m , the dimension is The minimum code distance is d = q m-1 (qr), is the set defined in Definition 2;

[0059] Definition 5: Based on the MPE definition of Definition 3, we define a q-ary RS code:

[0060]

[0061] in And α1,α2,…,α n is n mutually unequal elements;

[0062] Definition 6: Take m in Definition 4, including q module m The cyclotomic coset of -1 is defined as:

[0063] U λ = {q i λmod(q m -1)|1≤i≤m}

[0064] if Different, and but A subset of {λ1,λ2,…,λ s} is called q module m The complete set represented by the cyclotomic coset of -1.

[0065] Definition 7: Let m i is the value in definition 6 above. The period, m i Divides m and has subdomain set up yes A normal basis of , for every 1≤i≤s and 1≤j≤m in Definition 6 i , define the polynomial:

[0066]

[0067] Lemma 8: Take the polynomial defined in Definition 7 and let Fixed a group Normal basis {u1,u2,…,u m},make have:

[0068]

[0069] Proof of Lemma 8: First, according to Definition 7 and The normal basis {u1,u2,…,u m}have:

[0070]

[0071] Definition 9: Define from the above Lemma 8 arrive The mapping τ;

[0072] τ:

[0073] in represents the m-variable polynomial described in the second step of the proof of Lemma 8.

[0074] Lemma 10: The mapping defined in Definition 9, Actually A polynomial on .

[0075] Proof of Lemma 10:

[0076] It can be seen that the coefficients are all In addition,

[0077]

[0078]

[0079] Therefore, it is proved.

[0080] Definition 11: Let r be a positive integer satisfying:

[0081]

[0082] Assumptions are all modules q with weight at most r in Definition 6 m -1 cycloid coset, definition is the set in Lemma 8 A subset of is the polynomial defined in Definition 7, and

[0083] Definition 12: Definition in is the m-variable polynomial in Definition 9;

[0084] Lemma 13: Let is the set in Definition 12, then yes A set of bases and:

[0085]

[0086] Proof of Lemma 13: According to the definition, For q = 2, we have:

[0087]

[0088] For q>2:

[0089]

[0090] According to the proof process of Lemma 10 and Definition 12, we know that for Each polynomial in has an order at most r, according to The above conclusions can be obtained.

[0091] Definition 14: Let It is expressed as described by Definition 11 Zhang Cheng's Linear space is

[0092]

[0093] Lemma 15: For any vector Computation in the standard monomial basis exist It requires O(Km) operations. Similarly, the interpolation process is also in We can perform O(Km) operations in is the polynomial described in Definition 7.

[0094] Proof of Lemma 15: The result is given by the polynomial By definition, only if hour, There are non-zero monomials. Since Each There are at most m non-zero monomials. In addition, are disjoint, so the monomial set B i are also disjoint. Take the monomial basis B1, B2, ..., B t , in Then we have:

[0095]

[0096] in is the mole matrix, and is a finite field A set of normal bases. So A mi is reversible. Therefore, the matrix A = diag(A m1 ,A m2 ,…,A mt) and A -1 are all reversible. Given any Calculation in standard basis is to calculate a·A. It only takes O(Km) operations. Similarly, given a polynomial Assumptions in Yes a The coefficient vector of t So calculating a is calculating b·A -1 , this is It only takes O(Km) operations.

[0097] Definition 16: Let τ be the mapping in Definition 9. According to Lemma 13, for an m-variable polynomial F(x1,x2,…x m ) can be All The linear table gives the The univariate polynomial f(x) is

[0098]

[0099] Definition 17: Fixed a of Normal basis {u1,u2,…,u m}, defined in and One between Isomorphism:

[0100] π:

[0101] Corollary 18: Under the isomorphism π described in Definition 17, is the multi-point set defined in Definition 3, corresponding to Multi-point set π in -1 (P), and vice versa, according to the transformation defined in Lemma 8, we know that For an m-variable polynomial F(x1,x2,…x m ) is used for MPE. The MPE of a univariate polynomial f(x) is given by Definition 16, that is,

[0102]

[0103] Therefore, you can use Fast MPE algorithm for single variable polynomials to speed up MPE of polynomials in m variables.

[0104] Definition 19: Let is the set in Definition 12, Assumptions It is the base A polynomial under , and is a multi-point set defined in Definition 3, and satisfies

[0105] Theorem 20: According to the above Definition 19, under the isomorphism π described in Definition 17, if yes The multiplicative subgroup in , then F is in the point set The MPE can be Conducted Operation obtains, if N = O (q m ), then MPE and the interpolation of F can be On Calculated, if N is O(1) smooth (N = O(q m ), then MPE and the interpolation of F can be obtained in O((Km+NlogN)M q (m)) calculated, where M q (n) means The number of operations to multiply two univariate polynomials of degree less than n.

[0106] Proof of Theorem 20: Let τ be the mapping in Definition 9, and the MPE is calculated based on Definition 3. For any According to definition 9, is the corresponding univariate polynomial. According to Lemma 15, we can transform the MPE of F in P into f in π -1 (P) MPE. In the above example, both steps take O(Km) operations. Therefore, we only need to count the MPE of f(x) and the number of interpolation operations. yes The multiplicative subgroup in , then The non-zero polynomials of are:

[0107]

[0108] make but Compute using currently known integer modular arithmetic The cost is at most KO(logq m ) operations, according to the current research results on RS codes, Can be Most hits Calculated in operations, (n) represents The number of operations to multiply two univariate polynomials of degree less than n. calculate need If N = O(q m ), you can Expand to In this case Therefore, given can be correctly recovered. The interpolation process also maintains the same complexity as MPE. Furthermore, if N is O(1)-smooth, then and The interpolation can be done by The O(NlogN) FFT and inverse FFT operations in the calculation are used to prove this.

[0109] Theorem 21: According to the above definition 19, if where ω is The original element of The MPE can be On the contrary, if K≤N, given Then the recovery of F can be done in time complexity Next completed.

[0110] Proof of Theorem 21: Using the notation in the proof of Theorem 20, a univariate polynomial f(x) has at most K nonzero terms, but its order can reach q m -1, when K is much smaller than q m When , f(x) is a high-order sparse polynomial. Let S = {1,ω,ω 2 …,ω N-1}, based on previous research results, we use fast MPE and interpolation of sparse univariate polynomials, given f,ev S The MPE of (f) can be In In contrast, given the MPE ev S (f), if K≤N, then Conducted operations can uniquely recover f(x), plus O(M q (m)Km) of expenditure, thus it is proved.

[0111] Definition 22: Let is the information space, according to definition 12, we get There exists a mapping that converts the information Mapped into information polynomial F a , which is the encoding mapping:

[0112]

[0113] Theorem 23: RM q (r,m) is And has the following parameters:

[0114] Theorem 24: For the binary case, the largest a∈[0,2 m -1] is a=2 m-r +2 m-r+1 +…+2 m-1 =2 m -2 m-r , so for each f m Both have deg(f m ) <k=2 m -2 m-r +1;

[0115] Theorem 25: For the q-ary case, the Hamming weight wt q (a)≤rThe largest a∈[0,q m -1] is a=rq m-1 , so for each f m Both have deg(f m ) <k=rq m-1 +1;

[0116] Theorem 26: Let

[0117]

[0118] Therefore, D = N – k + 1 is RM q The minimum distance of RS(r,m) is also the minimum distance of RS(N,k).

[0119] Lemma 27: According to the work of Gao et al., unique decoding of RS codes requires Next, O(M q (n)logn) domain operation, where M q (n) means The number of operations of multiplying two univariate polynomials of order less than n in , if the set of evaluation points is The multiplication or addition subgroup of At most O(nlog 2 n).

[0120] Theorem 28: Let N = q mis the code length of the RM(r,m) code, and there is a decoding time complexity of O(Nlog 3+o(1) The only decoding algorithm for RM code of N) is:

[0121] Proof of Theorem 28: First, use the designed coding algorithm to encode the RM(r,m) code. Given the received vector The number of errors is less than half of the minimum distance, and assuming is the original codeword. First, we use r to input into the RS code decoder and calculate the correct message polynomial Next, using Lemma 13, The m-variable polynomial F = τ(f(x)) can be The above basis indicates that this process can be done in the domain The above can be obtained with at most O(Km) operations. According to Lemma 27, the unique decoding complexity in the domain The decoding complexity of RM code is (Km+Nlog 2+o(1) N)M q (m) = O(Nlog 3+o(1) N).

[0122] The present invention is implemented as follows: a scheme capable of encoding and decoding any r-order RM code with quasi-linear time complexity, aiming to optimize the time complexity of encoding and decoding any r-order RM code, and the unique decoding specifically includes the following steps (see Figure 1 ):

[0123] 1. Preprocessing stage: Define q-element RM according to Definition 4 q (r,m)RM code, defines the information according to Definition 7 and transforms it into a univariate polynomial.

[0124] 2. Coding stage:

[0125] Will The m-variable MPE problem on the extended domain is simplified to The univariate MPE problem on The MPE acceleration of the upper m-dimensional polynomial makes the coding complexity reach the quasi-linear level.

[0126] The encoding is performed using an RS code encoder.

[0127] 3. The receiving end obtains the unique decoding result stage:

[0128] The received codeword is input into the unique decoder of the RS code, the correct message polynomial is calculated, and the m-ary information polynomial is obtained through the mapping relationship, and the coefficients are the unique decoding results.

[0129] Through the above steps, the present invention is based on Multivariate Polynomial Rings and Extended Finite Fields on The idea of ​​transformation between univariate polynomial rings on The m-variable MPE problem on the extended domain is simplified to The univariate MPE problem on The RM code on the The RS code on the RM code is uniquely decoded, so that the encoding and unique decoding complexity of any r-order RM code reaches the quasi-linear level. This method is suitable for the industrial communication field that has high requirements on the time complexity of encoding and decoding, especially in the communication field.

[0130] The encoding method is an improved scheme capable of encoding any r-order RM code with quasi-linear time complexity.

[0131] Further, the The m-variable MPE problem on the extended domain is simplified to The univariate MPE problem on the RS code is encoded using the following steps:

[0132] Take the q-element RM in Definition 4 q The code length of (r,m)RM code is N = q m , the dimension is m i Based on Definition 7, the definition information is

[0133] Mapping information into information polynomials in Based on Definition 7;

[0134] Encoder calculation using RS code The returned c is the final encoded codeword.

[0135] Further, the The unique decoding of the RM code on The unique decoding of the RS code on the RS code is performed using the fast unique decoder of the RS code. The specific steps for unique decoding are as follows:

[0136] The received codeword Input to the RS code decoder to calculate the correct message polynomial

[0137] Calculated in The m-variable polynomial F = τ(f(x)).

[0138] The coefficient of the final output F is the unique decoding result.

[0139] based on Multivariate Polynomial Rings and Extended Finite Fields on Based on the idea of ​​transformation between univariate polynomial rings on , a quasi-linear time complexity encoding and decoding scheme for RM codes of arbitrary order is proposed. Two specific application embodiments of the embodiments of the present invention are:

[0140] Embodiment 1: Communication system

[0141] It is suitable for secure communications, such as military communications, complex channel transmission, etc. It is suitable for situations where the encoding and decoding speed is high.

[0142] Encoding stage: Information preprocessing: Defining q-element RM q (r,m)RM code, defines the information and converts it into a univariate information polynomial, and uses the encoder of the RS code to encode the information polynomial.

[0143] Transmission phase: the encoded information is transmitted through the channel.

[0144] Unique decoding stage: The received codeword is input into the unique decoder of the RS code, the correct message polynomial is calculated, and the m-ary information polynomial is obtained through the mapping relationship. The coefficients are the unique decoding results.

[0145] Embodiment 2: Video transmission system

[0146] Transmit video in a channel with additive noise interference, providing a reliable video transmission solution. It is suitable for situations where the encoding and decoding speed is high.

[0147] Encoding stage: Information preprocessing: Defining q-element RM q (r,m)RM code, defines the information and converts it into a univariate information polynomial, and uses the encoder of the RS code to encode the information polynomial.

[0148] Transmission phase: the encoded information is transmitted through the channel.

[0149] Unique decoding stage: The received codeword is input into the unique decoder of the RS code, the correct message polynomial is calculated, and the m-ary information polynomial is obtained through the mapping relationship. The coefficients are the unique decoding results.

[0150] The present invention mainly improves the following problems and defects of the prior art and achieves significant technical progress:

[0151] RM codes are slow to decode: no algorithm has been proposed earlier that can achieve quasi-linear decoding complexity for RM codes of arbitrary order.

[0152] Limited applicability: Due to the disadvantage of slow decoding speed of RM code, it is not applicable in many complex channels and scenarios with high requirements on encoding and decoding time complexity.

[0153] In view of the problems existing in the prior art, the unique decoding technical solution adopted by the present invention is:

[0154] Step 1: Take the q-element RM in Definition 4 q The code length of (r,m)RM code is N = q m , the dimension is m i Based on Definition 7, the definition information is

[0155] Step 2: Map the information into an information polynomial in Based on Definition 7.

[0156] Step 3: Calculate using the RS code encoder The returned c is the final encoded codeword.

[0157] Step 4: Receive codeword Input to the RS code decoder to calculate the correct message polynomial

[0158] Step 5: Calculate The m-variable polynomial F=τ(f(x)), and the coefficient of the final output F is the unique decoding result.

[0159] Step 6: Analyze the time complexity of the encoding and decoding scheme, as follows:

[0160] 6.1 The coding scheme, RM q The encoding of (r,m) needs to be In terms of time complexity (NlogNM q (m)) = O(Nlog 2+o(1) N) is completed.

[0161] The unique decoding scheme described in 6.2 can be proved by the proof of Theorem 28, let N = q m is the code length of the RM(r,m) code, and there is a decoding time complexity of O(Nlog 3+o(1) N)'s unique decoding algorithm for RM code.

[0162] The present invention provides a quasi-linear time complexity encoding and decoding algorithm method for arbitrary-order RM codes. Multivariate Polynomial Rings and Extended Finite Fields on The idea is to transform between univariate polynomial rings on . Here is how the method works in detail:

[0163] Coding process:

[0164] 1) Information preprocessing:

[0165] Define q-ary RM q (r,m)RM code, defines information and converts it into a univariate information polynomial

[0166] 2) Encode using the RS code encoder:

[0167] The information polynomial is encoded using an RS code encoder.

[0168] Will The m-variable MPE problem on the extended domain is simplified to The univariate MPE problem on is the key to achieving quasi-linear coding complexity.

[0169] Unique decoding process:

[0170] 1) Decode using the unique decoder of RS code:

[0171] The received codeword is input into the unique decoder of the RS code to calculate the correct message polynomial.

[0172] 2) Calculate the final decoding result:

[0173] The m-dimensional information polynomial is obtained through the mapping relationship, and the coefficient is the unique decoding result.

[0174] Will The RM code on the The decoding of RS codes on is the key to making the unique decoding complexity of any r-order RM code reach the quasi-linear level.

[0175] Technical advantages:

[0176] The technical solution of the present invention fills the technical gap in the industry at home and abroad: As shown in Table 1, before this, the decoding complexity when q=2 is O(Nlog r N), the decoding complexity when q>2 is O(q 2 Npoly(logN)), there is still much room for optimization in the field of quasi-linear time algorithms for decoding arbitrary-order RM codes. Multivariate Polynomial Rings and Extended Finite Fields on The idea of ​​transforming between univariate polynomial rings on , so that the encoding and decoding complexity of any r-order RM code reaches the quasi-linear level. Specifically, the decoding complexity optimization when q = 2 reaches O(Nlog 2+o(1) N), the decoding complexity for any q is O(Nlog 3+o(1) N), achieving significant technological advancement.

[0177] Table 1

[0178]

[0179] Accelerated RM code encoding and decoding: The present invention is based on Multivariate Polynomial Rings and Extended Finite Fields on The idea of ​​transforming between univariate polynomial rings on the RM code makes the encoding and decoding complexity reach the quasi-linear level, speeding up the encoding and decoding time.

[0180] Wide applicability: The proposed encoding and decoding method is suitable for the industrial communication field with high requirements on the encoding and decoding time complexity.

[0181] Strong adaptability: The present invention is applicable to RM codes of various parameters and various application scenarios.

[0182] Technological innovation: The encoding and decoding method of the present invention is an effective supplement and improvement to the existing RM code encoding and decoding technology, demonstrating technological innovation in the field of communications.

Claims

1. A quasi-linear time complexity encoding and unique decoding algorithm for arbitrary order Reed-Muller codes, characterized in that: include: Define q-ary RMq(r, m)RM code, define information and transform it into univariate information polynomial; Will The m-variable MPE problem on the extended domain is simplified to The univariate MPE problem on , using the RS code encoder to encode, so as to achieve The MPE acceleration of the upper m-dimensional polynomial makes the coding complexity reach the quasi-linear level; The received codeword is input into the unique decoder of the RS code, the correct message polynomial is calculated, and the m-ary information polynomial is obtained through the mapping relationship, and the coefficients are the unique decoding results.

2. The quasi-linear time complexity encoding and unique decoding algorithm for arbitrary order Reed-Muller codes as claimed in claim 1, characterized in that: Use the RS code encoder to encode, so as to achieve The MPE acceleration of the m-dimensional polynomial makes the encoding complexity reach the quasi-linear level. The specific steps are as follows: Take the q-ary RM in Definition 4 q The code length N of the (r, m)RM code is q m , the dimension is m i Based on Definition 7, the definition information is Mapping information into information polynomials in Based on Definition 7; Encoder calculation using RS code The returned c is the final encoded codeword.

3. The quasi-linear time complexity encoding and unique decoding algorithm for arbitrary order Reed-Muller codes as claimed in claim 2, characterized in that: Definition 4: Based on the MPE definition of Definition 3, a q-ary RM code is defined as: Its code length is N = q m , the dimension is The minimum code distance is d = q m-1 (qr), is the set defined in Definition 2.

4. The quasi-linear time complexity encoding and unique decoding algorithm for arbitrary order Reed-Muller codes as claimed in claim 3, characterized in that: Definition 3: Multivariable multi-point assignment (hereinafter referred to as MPE) is defined as follows: Given N assignment point sets Take the polynomial F in Definition 1 and the set in Definition 2 And calculate (F(a1),…,F(a N )):

5. The quasi-linear time complexity encoding and unique decoding algorithm for arbitrary order Reed-Muller codes as claimed in claim 3, characterized in that: Definition 1: Define a finite field polynomials of m variables on the Define the total degree of F as:

6. The quasi-linear time complexity encoding and unique decoding algorithm for arbitrary order Reed-Muller codes as claimed in claim 3, characterized in that: Definition 2: Definition set are all polynomials with a total degree less than or equal to r subspace, the dimension is 7. The quasi-linear time complexity encoding and unique decoding algorithm for arbitrary order Reed-Muller codes as claimed in claim 1, characterized in that: Definition 7: Let m i is the value in definition 6 above. The period, m i Divides m and has subdomain set up yes A normal basis of , for every 1≤i≤s and 1≤j≤m in Definition 6 i , define the polynomial:

8. The quasi-linear time complexity encoding and unique decoding algorithm for arbitrary order Reed-Muller codes as claimed in claim 7, characterized in that: Definition 6: Take m in Definition 4, including q module m The cyclotomic coset of -1 is defined as: U λ ={q i λmod(q m -1)|1≤i≤m} if Different, and but A subset of {λ1,λ2,…,λ s } is called q module m The complete set represented by the cyclotomic coset of -1.

9. The quasi-linear time complexity encoding and unique decoding algorithm for arbitrary order Reed-Muller codes as claimed in claim 1, characterized in that: Will The RM code on the The unique decoding of RS codes on , so that the unique decoding complexity of any r-order RM code reaches the quasi-linear level, the specific steps are as follows: The received codeword Input to the RS code decoder to calculate the correct message polynomial Calculated in The m-variable polynomial F = τ(f(x)); The coefficient of the final output F is the unique decoding result.

10. An application of the quasi-linear time complexity encoding and unique decoding algorithm of arbitrary order Reed-Muller codes according to any one of claims 1 to 9, characterized in that: It should be applied in technical fields including military communications and complex channel transmission.

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