Any-order Reed-Muller code quasi-linear time complexity coding and decoding method and system

Abstract

The invention belongs to the technical field of communication, and discloses a quasi-linear time complexity coding and list decoding algorithm for any-order RM codes, which comprises the following steps of: defining q-element RMq (r, m) RM codes, defining information, converting the information into a unary polynomial, simplifying an m variable MPE problem on # imgabs0 # into a single variable MPE problem on # imgabs1 # of an extension field, and decoding the single variable MPE problem on the # imgabs1 # of the extension field. The method comprises the following steps of: performing MPE acceleration on an m-element polynomial on # imgabs2 # so as to realize MPE acceleration on an m-element polynomial on # imgabs2 #, so that the coding complexity reaches a quasi-linear level, converting list decoding of an RM code on # imgabs3 # into list decoding of an RS code on # imgabs4 # by a receiving end, and calculating and selecting a decoding result and a received code word, so that the list decoding complexity of any r-order RM codes reaches a quasi-linear level of O (Nlog3 + o (1) N). According to the method, a quasi-linear time algorithm of any r-order RM code list decoding is realized on the basis of a transformation idea between a multivariate polynomial ring on # imgabs5 # and a univariate polynomial ring on an extended finite field # imgabs6 #, RM code coding and list decoding are accelerated, and the applicability of a coding and decoding 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 list decoding method and system for arbitrary order Reed-Muller (hereinafter referred to as RM) codes. 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 the number of errors e in the communication channel is assumed to be less than half of the minimum distance of the code, the original codeword can be uniquely recovered, which is a unique decoding. If the number of errors e is greater than half of the minimum distance, list decoding can be used, which receives a vector and outputs a list of codewords within its Hamming distance e. For any error fraction 0≤ρ≤1 and integer l>1, a linear code It is called (ρ,l)-list decodable if for every accepted codeword The number of codewords whose Hamming distance does not exceed ρN does not exceed l.

[0004] RM codes were 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 codes is actually multi-variable multi-point assignment (hereinafter referred to as MPE). Designing an approximately linear time algorithm for the input size of MPE is a daunting challenge. In current research, almost all known quasi-linear list decoding algorithms for RM codes are designed for parameter regions with constant order r, and most list decoding algorithms can only achieve quasi-linear time decoding complexity when the order r is a constant.

[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. Before this, although researchers have made great efforts, the quasi-linear time algorithm for arbitrary-order RM codes has not yet been discovered, 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 RM code encoding and list decoding in the background technology, the present invention proposes a method that can implement encoding and list decoding of any r-order RM code with quasi-linear time complexity.

[0007] 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 list on the decoded into The Reed-Solomon (hereinafter referred to as RS) code list decoding on.

[0008] The proof of this algorithm requires some definitions and theorems as follows:

[0009] Definition 1: Define a finite field polynomials of m variables on the Define the total degree of F as:

[0010] Definition 2: Define a set are all polynomials with a total degree less than or equal to r subspace, the dimension is

[0011] 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 )):

[0012]

[0013] Definition 4: Based on the MPE definition of Definition 3, we define a q-ary RM code:

[0014]

[0015] 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;

[0016] Definition 5: Based on the MPE definition of Definition 3, we define a q-ary RS code:

[0017]

[0018] in And α1,α2,…,α n is n mutually unequal elements;

[0019] Definition 6: Take m in Definition 4, including q module m The cyclotomic coset of -1 is defined as:

[0020] U λ = {q i λmod(q m -1)|1≤i≤m}

[0021] 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.

[0022] 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] Lemma 8: Take the polynomial defined in Definition 7 and let Fixed a group Normal basis {u1,u2,…,u m},make have:

[0025]

[0026] Proof of Lemma 8: First, according to Definition 7 and The normal basis {u1,u2,…,u m}have:

[0027]

[0028] Definition 9: Define the following from Lemma 8 above: arrive The mapping τ

[0029] τ:

[0030] in represents the m-variable polynomial described in the second step of the proof of Lemma 8.

[0031] Lemma 10: The mapping defined in Definition 9, Actually A polynomial on .

[0032] Proof of Lemma 10:

[0033] It can be seen that the coefficients are all In addition,

[0034]

[0035] Therefore, it is proved.

[0036] Definition 11: Let r be a positive integer satisfying:

[0037]

[0038] 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

[0039] Definition 12: Definition in is the m-variable polynomial in Definition 9;

[0040] Lemma 13: Let is the set in Definition 12, then yes A set of bases and:

[0041]

[0042] Proof of Lemma 13: According to the definition, For q = 2, we have:

[0043]

[0044] For q>2:

[0045]

[0046] 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.

[0047] Definition 14: Let It is expressed as described by Definition 11 Zhang Cheng's Linear space is

[0048]

[0049] 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.

[0050] 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:

[0051]

[0052] 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.

[0053] 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

[0054]

[0055] Definition 17: Fixed a of Normal basis {u1,u2,…,u m}, defined in and One between Isomorphism:

[0056] π:

[0057] 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,

[0058]

[0059] Therefore, you can use Fast MPE algorithm for single variable polynomials to speed up MPE of polynomials in m variables.

[0060] 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

[0061] 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.

[0062] 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:

[0063]

[0064] 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.

[0065] 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.

[0066] 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.

[0067] 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:

[0068]

[0069] Theorem 23: RM q (r,m) is And has the following parameters:

[0070] 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;

[0071] 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;

[0072] Theorem 26: Let

[0073]

[0074] Therefore, D = N – k + 1 is RM q The minimum distance of RS(r,m) is also the minimum distance of RS(N,k).

[0075] Lemma 27: According to the existing research conclusions, for a q-ary [n,k] RS code RS(n,k) is (ρ,l)-list decodable, its decoding radius ρ and list size l are:

[0076] l=O(1 / ∈)

[0077] Where ∈>0 represents a very small positive number, is the relative minimum distance, which can be found in O(∈ -ω- 1 nlog 2 nloglogn) complexity, where ω represents the exponent of matrix multiplication.

[0078] The present invention is implemented as follows: a method for encoding and list decoding any r-order RM code with quasi-linear time complexity is intended to optimize the time complexity of encoding and list decoding of any r-order RM code, and list decoding specifically includes the following steps:

[0079] 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.

[0080] 2. Coding stage:

[0081] 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.

[0082] The encoding is performed using an RS code encoder.

[0083] 3. Receiver list decoding list stage:

[0084] Will The list decoding of the RM code on is converted into The decoding result and the received codeword are calculated and selected, so that the decoding complexity of any r-order RM code list reaches O(Nlog 3+o(1) N) of the quasi-linear level.

[0085] 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 list decoding of the RM code on is converted into The list decoding of RS codes on , thus making the encoding and list decoding complexity of any r-order RM code reach the quasi-linear level. This method is suitable for industrial communication fields that require high encoding and decoding time complexity or application scenarios of complex channels that require high error correction capabilities, especially in the communication field.

[0086] The encoding method is an improved scheme capable of encoding any r-order RM code with quasi-linear time complexity.

[0087] 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:

[0088] Take the q-element RM in Definition 4 q The code length of (r,m)RM code is N = q m , the dimension is mi Based on Definition 7, the definition information is

[0089] Mapping information into information polynomials in Based on Definition 7;

[0090] Encoder calculation using RS code The returned c is the final encoded codeword.

[0091] Further, the The list decoding of the RM code on is converted into The specific steps of list decoding of RS code on the RS code list decoding using the RS code fast list decoder are as follows:

[0092] Receive codeword And order

[0093] Use the existing fast RS list decoder (ρ, l) to decode the received codeword r, and let L ′ is its output;

[0094] Traversal According to Definition 3, we calculate Recovery information F is the m-variable polynomial in Definition 1, let L = L∪a, until the traversal ends;

[0095] Output d H represents the Hamming distance, and the list size is at most l;

[0096] The final output L is the decoding list.

[0097] 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:

[0098] First, the present invention is based on Multivariate Polynomial Rings and Extended Finite Fields on The key advantages of this approach include:

[0099] 1. Enhanced error detection and correction capabilities: The list decoding algorithm RM q The (r,m) list decoding radius reaches the Johnson bound, making this method robust to transmission errors exceeding (d is RM q When the minimum code distance (r, m) is reached, it can effectively detect errors and output a decoding list, thereby increasing the error detection and correction capabilities.

[0100] 2. 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 list decoding of the RM code on is converted into The list decoding of the RS code on the RM code makes the complexity of RM code encoding and list decoding reach a quasi-linear level, speeding up the encoding and decoding.

[0101] 3. A quasi-linear time algorithm for decoding arbitrary r-order RM code lists is implemented: 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 list on the decoded into The RS code list decoding on is realized, thus realizing a quasi-linear time algorithm for decoding any r-order RM code list.

[0102] 4. Wide applicability: This method is suitable for industrial communication fields that require high encoding and decoding time complexity or various application scenarios of complex channels that require high error correction capabilities, including but not limited to secure communications, sensitive information transmission, and steganography.

[0103] The present invention not only represents a significant technical advancement, but also provides an effective improvement and supplement to the existing RM code encoding and decoding. By implementing this method, RM code encoding and list decoding can be accelerated, error detection and correction capabilities can be enhanced, and the growing demand for secure communication and coping with complex channels can be met.

[0104] Second, the present invention is based on Multivariate Polynomial Rings and Extended Finite Fields on The idea of ​​transforming between univariate polynomial rings on the code not only effectively improves the error detection and correction capabilities of the previous RM code, but also accelerates the RM code encoding and list decoding, achieving significant technological progress.

[0105] Improve the error detection and correction capabilities of the previous RM code: the list decoding algorithm RM q The decoding radius of the (r, m) list reaches the Johnson bound, and the present invention improves the error detection and correction capability of the RM code.

[0106] Accelerating RM code encoding and decoding: This method reduces the encoding and list decoding time complexity to a quasi-linear level, thereby speeding up encoding and decoding.

[0107] A quasi-linear time algorithm for decoding a list of RM codes of arbitrary order is implemented: This algorithm implements a quasi-linear time algorithm for decoding a list of RM codes of arbitrary order.

[0108] Strong adaptability: The present invention is applicable to RM codes of various parameters and various application scenarios.

[0109] 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.

[0110] Third, the expected benefits and commercial value of the technical solution of the present invention after transformation are:

[0111] Encode the information through RM code, place it in a complex channel, and the receiver decodes the RM code list. This operation can improve the error detection and correction capabilities of the previous RM code and accelerate the encoding and decoding of the RM code. This technology can be used in secure communications, sensitive information transmission, steganography and other fields.

[0112] The technical solution of the present invention fills the technical gap in the industry at home and abroad: before this, the quasi-linear time algorithm for decoding arbitrary-order RM code lists has not been discovered. To solve this problem, 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 list decoding of the RM code on is converted into The list decoding of RS codes on , so that the encoding and list decoding complexity of any r-order RM code reaches the quasi-linear level.

[0113] The technical solution of the present invention solves a technical problem that people have been eager to solve but have never succeeded in solving: the quasi-linear time algorithm for traditional arbitrary-order RM codes has not yet been discovered, while the method realizes a quasi-linear time algorithm for decoding a list of arbitrary-order RM codes. BRIEF DESCRIPTION OF THE DRAWINGS

[0114] Figure 1 The flowchart of the quasi-linear time complexity encoding and list decoding algorithm of arbitrary-order RM codes of the present invention is shown in FIG. DETAILED DESCRIPTION

[0115] 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.

[0116] In view of the problems existing in the prior art, the present invention provides a quasi-linear time algorithm for implementing arbitrary-order RM code encoding and list decoding. The present invention is described in detail below.

[0117] The present invention uses the Multivariate Polynomial Rings and Extended Finite Fields on The idea of ​​transformation between univariate polynomial rings on , so that the encoding and list decoding complexity of any r-order RM code reaches the quasi-linear level. The list decoding algorithm specifically includes the following steps:

[0118] 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.

[0119] 2. Coding stage:

[0120] 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.

[0121] The encoding is performed using an RS code encoder.

[0122] 3. The receiving end obtains the decoding list stage:

[0123] Will The list decoding of the RM code on is converted into The list decoding of RS codes on , thus making the list decoding complexity of any r-order RM code reach the quasi-linear level.

[0124] The received information is decoded using a fast RS list decoder (ρ, e), and the decoding list is determined by the decoding result of the RS decoder and the received codeword.

[0125] 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 list decoding of the RM code on is converted into The list decoding of RS codes on , thus making the encoding and list decoding complexity of any r-order RM code reach the quasi-linear level. This method is suitable for industrial communication fields that require high encoding and decoding time complexity or application scenarios of complex channels that require high error correction capabilities, especially in the communication field.

[0126] 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 list decoding scheme for arbitrary order RM codes is proposed. Two specific application embodiments of the embodiments of the present invention are:

[0127] Embodiment 1: Communication system

[0128] Suitable for secure communications, such as military communications, complex channel transmission, etc. Suitable for channels with strong anti-interference capabilities.

[0129] 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.

[0130] Transmission phase: the encoded information is transmitted through the channel.

[0131] List decoding stage: Decode the received codeword using the existing fast RS list decoder (ρ, l), use the decoding result of the RS decoder and the received codeword for calculation and selection, and finally output a decoding list with a list size of at most l;

[0132] Embodiment 2: Video transmission system

[0133] Transmit video in channels with additive noise interference, providing a reliable video transmission solution. Suitable for channels with strong anti-interference capabilities.

[0134] Encoding stage: Information preprocessing: convert video information into pixel values, convert pixel values ​​into information sequences in order according to requirements, and define q-element RM q (r,m)RM code, converts the above information sequence into a univariate information polynomial, and uses the encoder of RS code to encode the information polynomial.

[0135] Transmission phase: The encoded pixel values ​​are passed through the channel.

[0136] List decoding stage: The received codeword is decoded using the existing fast RS list decoder (ρ, l), and the decoding result of the RS decoder is used with the received codeword for calculation and selection, and finally a decoding list with a list size of at most l is output, and the original video is reconstructed according to the decoding result;

[0137] The present invention mainly improves the following problems and defects of the prior art and achieves significant technical progress:

[0138] Weak error detection and correction capabilities: The only decoding algorithm is unable to detect or correct errors when there are more than (d is RM q When the minimum code distance (r, m) is reached, it is not possible to effectively detect and correct errors.

[0139] RM codes are slow to decode: no algorithm has been proposed earlier that can achieve quasi-linear decoding complexity for RM code lists of arbitrary order.

[0140] Limited applicability: Due to its weak error detection and correction capabilities and slow decoding speed, RM code is not applicable in many complex channels and scenarios with high requirements for encoding and decoding time complexity.

[0141] In view of the problems existing in the prior art, the technical solution of list decoding adopted by the present invention is:

[0142] 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

[0143] Step 2: Map the information into an information polynomial in Based on Definition 7;

[0144] Step 3: Calculate using the RS code encoder Return c is the final encoded codeword;

[0145] Step 4: Receive codeword And order

[0146] Step 5: Use the existing fast RS list decoder (ρ, l) to decode the received codeword r, and let L ′ is its output;

[0147] Step 6: Traverse According to Definition 3, we calculate Recovery information F is the m-variable polynomial in Definition 1, let L = r∪a, until the traversal ends;

[0148] Step 7. Output d H represents the Hamming distance, and the size of the list is at most l, and the final output L is the decoding list;

[0149] Step 8: Analyze the time complexity of the encoding and list decoding scheme, as follows:

[0150] 8.1 The coding scheme, RM q The encoding of (r,m) is the entire point set exist Multi-point polynomial assignment on, according to Theorem 20, we know that the encoding process is Time complexity Completed. If q m -1 is O(1) smoothing, then Can be In terms of time complexity (NlogNM q (m)) = O(Nlog 2+o(1) N) is completed, and F is calculated separately a (O) It is done in time complexity O(1), so in this case, 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.

[0151] 8.2 List decoding scheme, let N = q m is the code length of the RM(r,m) code, assuming that the RS list decoder (ρ,l) is a fast list decoding algorithm for RS codes. By Lemma 27, for both binary and q-ary RM codes List translatable, where ∈>0 and the decoding complexity is

[0152] More specifically, for q-ary RM codes, RM q The Johnson bound on the decoding radius of the (r,m) list is

[0153]

[0154] Therefore, RM q The list decoding radius of (r,m) reaches the Johnson bound.

[0155] In summary, the list decoding algorithm is effective for q-ary RM code RM q (r,m), There is a decoding time complexity of And reach the Johnson radius List decoding algorithm, where δ = 1-r / q is RM q The relative minimum distance of (r,m), ∈>0 represents a very small positive number, ω represents the exponent of matrix multiplication, for the binary RM code RM2(r,m), there is a decoding time complexity of And reach the Johnson radius List decoding algorithm;

[0156] The present invention provides a quasi-linear time complexity encoding and list 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:

[0157] Coding process:

[0158] 1) Information preprocessing:

[0159] Define q-ary RM q (r,m)RM code, defines information and converts it into a univariate information polynomial

[0160] 2) Encode using the RS code encoder:

[0161] The information polynomial is encoded using an RS code encoder.

[0162] 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.

[0163] List decoding process:

[0164] 1) Use the fast list decoder of RS code to perform list decoding:

[0165] The received codeword is decoded using the existing fast RS list decoder (ρ, l).

[0166] 2) Calculate the final list:

[0167] The decoding result of the RS list decoder is used together with the received codeword for calculation and selection, and finally a decoding list with a list size of at most l is output.

[0168] Will The RM code list on the decoded into The decoding of RS code lists on is the key to making the decoding complexity of any r-order RM code list reach the quasi-linear level.

[0169] The following are two specific industrial application embodiments of the above invention:

[0170] Example 1: Data transmission and error correction in satellite communications

[0171] In satellite communication systems, the reliability of data transmission is crucial. Due to the long distance and complex environment of satellite communication, data is easily affected by factors such as noise and interference during transmission, resulting in an increase in the bit error rate. In order to improve the reliability of data transmission, efficient coding and error correction technology is usually required.

[0172] The combined system based on Reed-Muller code (RM code) and Reed-Solomon code (RS code) proposed by the present invention can realize efficient encoding and list decoding in satellite communication. Specifically, before data transmission, the data is encoded using RM code, which has a strong error correction capability and is suitable for processing random errors in long-distance transmission. Subsequently, RS code is used for further encoding, which increases the ability to handle burst errors. At the receiving end, the fast list decoding algorithm of the present invention can be used to perform efficient decoding with quasi-linear time complexity, quickly restore the original data, and correct errors generated during the transmission process. This combined encoding and decoding strategy can significantly improve the data transmission reliability and efficiency of satellite communication systems, reduce bit error rates, and ensure data integrity and accuracy.

[0173] Example 2: Signal processing in deep space probes

[0174] In deep space exploration missions, the communication distance between the probe and the ground station is extremely long, the signal strength is weak, and it is often interfered by various external factors such as cosmic rays and solar wind, which makes it easy for a large number of errors to occur during data transmission. In order to ensure that the important scientific data collected by the probe can be accurately transmitted back to the earth, advanced encoding and decoding technology must be used to enhance the signal's anti-interference ability.

[0175] The RM code and RS code combination system of the present invention can be applied to the communication system of deep space probes. At the data sending end, the probe first encodes the collected scientific data by RM code, and then encodes it by RS code. This combined coding scheme can provide dual protection for random errors and continuous burst errors within a single data block. At the receiving end, the ground station decodes and corrects errors of the received signals in an efficient manner through the fast list decoding system of the present invention to ensure the reliability and accuracy of the data. Since signal processing in deep space exploration needs to be performed as efficiently as possible under limited computing resources, the algorithm with quasi-linear time complexity proposed in the present invention can meet the needs of efficient decoding in deep space exploration missions and significantly improve the robustness and reliability of communication links.

[0176] The technical solution of the present invention fills the technical gap in the industry at home and abroad: As shown in Table 1, the first four rows are the relevant results of the existing RM code list decoding, and the last two rows are the results of this work. It can be seen that excluding the results of this work, the quasi-linear time algorithm for decoding any r-order RM code list has not yet been discovered. The operation on the r (N 3 ). To solve this problem, the present invention is based on Multivariate Polynomial Rings and Extended Finite Fields on The idea of ​​transforming between univariate polynomial rings on the same RM code is to make the encoding and list decoding complexity of any r-order RM code reach O(Nlog 3+o(1) N), which accelerates the list decoding of RM codes. At the same time, it can be seen that the decoding radius of list decoding reaches the Johnson bound. Compared with unique decoding, the error detection and correction capability of RM is improved.

[0177] 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 complexity of encoding and list decoding reach O(Nlog 3+o(1) N) quasi-linear level, speeding up encoding and decoding.

[0178] Wide applicability: The proposed coding and decoding method is suitable for industrial communication fields with high requirements on the time complexity of coding and decoding or various application scenarios of complex channels that require high error correction capabilities.

[0179] Improve the error detection and correction capabilities of the previous RM code: the list decoding algorithm RM q The decoding radius of the (r, m) list reaches the Johnson bound, and the present invention improves the error detection and correction capability of the RM code.

[0180] Strong adaptability: The present invention is applicable to RM codes of various parameters and various application scenarios.

[0181] 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.

[0182] Table 1. Comparison of decoding results of RM code list

[0183]

[0184] It should be noted that the embodiments of the present invention can be implemented by hardware, software, or a combination of software and hardware. The hardware part can be implemented using dedicated logic; the software part can be stored in a memory and executed by an appropriate instruction execution system, such as a microprocessor or dedicated design hardware. It can be understood by a person of ordinary skill in the art that the above-mentioned devices and methods can be implemented using computer executable instructions and / or contained in a processor control code, such as a carrier medium such as a disk, CD or DVD-ROM, a programmable memory such as a read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier. Such code is provided on the carrier medium. The device and its modules of the present invention can be implemented by hardware circuits such as very large-scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, etc., or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., can also be implemented by software executed by various types of processors, and can also be implemented by a combination of the above-mentioned hardware circuits and software, such as firmware.

[0185] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any modifications, equivalent substitutions and improvements made by any technician familiar with the technical field within the technical scope disclosed by the present invention and within the spirit and principle of the present invention should be covered by the protection scope of the present invention.

Claims

1. A method for encoding and list decoding any r-order RM code with quasi-linear time complexity, characterized in that: include: Define a q-ary Reed-Muller code (RM code) RMq(r, m), and map the input information into a univariate information polynomial; Will The m-variable MPE (polynomial evaluation) problem on the extended domain The univariate MPE problem on ; Using RS code (Reed-Solomo n The encoder of the code) encodes the simplified univariate MPE problem to generate the encoded codeword c, so that the encoding complexity reaches the quasi-linear level.

2. The encoding and decoding method for encoding and list decoding any r-order RM code with quasi-linear time complexity as claimed in claim 1, characterized in that: The method further comprises the following steps: Receive input codeword And initialize the list L to an empty set; Transform the list decoding problem of RM code into The list decoding problem of RS codes on , uses a fast RS list decoder to decode the received codeword r and generate a candidate list L ′; Traverse each codeword c in L′, calculate and recover the information based on the predefined m-variable polynomial Add the qualified a to the list L; Output the final decoding list that meets the Hamming distance condition where d H represents the Hamming distance, and ρ is the predetermined distance threshold.

3. The encoding and decoding method for encoding and list decoding any r-order RM code with quasi-linear time complexity as claimed in claim 1, characterized in that: 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 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.

4. The coding and decoding method for encoding and list decoding any r-order RM code with quasi-linear time complexity as claimed in claim 1, characterized in that Will The RM code list on the decoded into The list decoding of RS codes on , so that the list decoding complexity of any r-order RM code reaches O(Nlog 3+o(1) N) quasi-linear level, the specific steps are as follows: Receive codeword And order Decode the received codeword r using the existing fast RS list decoder (ρ, l), and let L′ be its output; Traversal According to Definition 3, we calculate Recovery information F is the m-variable polynomial in Definition 1, let L = L∪ a, until the traversal ends; Output d H represents the Hamming distance, and the list size is at most l; The final output L is the decoding list.

5. A coding and decoding system based on Reed-Muller code and Reed-Solomon code, characterized in that: The system includes: A definition and mapping module, used to define a q-ary Reed-Muller code (RM code) RM_q(r, m) and map input information into a univariate information polynomial; Polynomial simplification module, used to reduce the m-variable polynomial evaluation (MPE) problem on F_q to a univariate MPE problem on the extended domain F_q^m; An encoder module, used for encoding the simplified univariate MPE problem using an encoder of a Reed-Solomon code (RS code) to generate an encoded codeword; The fast list decoding module is used to receive the codeword and decode it through the fast RS list decoder, and output a decoding list that meets the predetermined conditions.

6. The encoding and decoding system as claimed in claim 5, characterized in that The definition and mapping module is also used to calculate and generate the code length N=q^m and dimension K of the q-ary Reed-Muller code.

7. The encoding and decoding system as claimed in claim 5, characterized in that The fast list decoding module receives an input codeword and initializes the list to an empty set, then transforms the list decoding problem of the RM code into the list decoding problem of the RS code on F_(q^m), and calls the fast RS list decoder to generate a candidate list.

8. The encoding and decoding system as claimed in claim 5, characterized in that The fast list decoding module further includes a recovery module for traversing each codeword in the candidate list, calculating and recovering information based on a predefined m-variable polynomial, and outputting a final decoding list that meets the Hamming distance condition.

Images