Algebraic Decoding Research Articles

Efficient Low-Complexity Decoding of CCSDS Reed–Solomon Codes Based on Justesen’s Concatenation

Forward error correction (FEC) is a key capability in modern satellite communications that provide the system designer with the needed flexibility to comply with the different applications' requirements. Reed-Solomon (RS) codes are well known for their ability to optimize between the system power, bandwidth, data rate, and the quality of service. This paper introduces an efficient decoding scheme for decoding the RS codes adhering to the Consultative Committee for Space Data Systems (CCSDS) standards based on Justesen's construction of concatenation. To maintain the standard output size, the proposed scheme first encodes every m - 1 bits using the single-parity-check (SPC) code, while the RS code encodes K SPC codewords into N symbols that are of the same size as CCSDS standard. Decoding on the inner SPC code is based on maximum-likelihood decoding Kaneko algorithm, while for the proposed coding scheme, the reduced test-pattern Chase algorithm is adapted for decoding the outer RS code. The simulation results show the coding gains of 1.4 and 7 dB compared with the algebraic decoding of RS codes over the AWGN and Rayleigh fading channels, respectively. Moreover, the adopted reduced test-pattern Chase algorithm for decoding the RS code achieves an overall complexity reduction of 40% compared with the conventional Chase decoding algorithm.

An efficient combination between Berlekamp-Massey and Hartmann Rudolph algorithms to decode BCH codes

In digital communication and storage systems, the exchange of data is achieved using a communication channel which is not completely reliable. Therefore, detection and correction of possible errors are required by adding redundant bits to information data. Several algebraic and heuristic decoders were designed to detect and correct errors. The Hartmann Rudolph (HR) algorithm enables to decode a sequence symbol by symbol. The HR algorithm has a high complexity, that's why we suggest using it partially with the algebraic hard decision decoder Berlekamp-Massey (BM). In this work, we propose a concatenation of Partial Hartmann Rudolph (PHR) algorithm and Berlekamp-Massey decoder to decode BCH (Bose-Chaudhuri-Hocquenghem) codes. Very satisfying results are obtained. For example, we have used only 0.54% of the dual space size for the BCH code (63,39,9) while maintaining very good decoding quality. To judge our results, we compare them with other decoders.

INCREASE IN EFFICIENCY OF DECODING OF CODES OF READ-SOLOMON ON THE GENERALIZED MINIMUM DISTANCE

In the modern systems of transfer and storage of information for correction of the arising mistakes noiseproof codes of Read-Solomon widely are used. With use of soft decisions apply decoding of these codes on the generalized minimum distance which advantage is simplicity of realization to correction of mistakes. In work the algorithm of decoding of codes of Read-Solomon on the generalized minimum distance which feature is use of the algebraic decoder correcting errors abroad a half of the minimum code distance with use of soft decisions is offered. The algebraic decoder realizes syndromic decoding and is based on application of analytical continuation of an algorithm of Berlekempa-Messi for 2τ iterations (τ-number of in addition corrected wrong symbols). He provides search of positions of tC+τ of wrong symbols in a code word (tC - number of the wrong symbols which are guaranteed corrected by a code) which locators would be the return to roots of a possible polynom of locators of errors of degree tC + τ. Search of positions of mistakes is carried out in ascending order of nadezhnost of symbols of the accepted code word. The efficiency of correction of mistakes was investigated by the offered algorithm in the channel with additive white Gaussian noise by imitating modeling on the COMPUTER. Researches were conducted for Read-Solomon's codes defined over the field of GF(28). The additional code prize provided with an algorithm at correction on iteration of three additional mistakes in relation to Read-Solomon (255,239,17) code reaches 0,26 dB. The additional code prize for Read-Solomon (255,127,129) code at correction on iteration of two additional mistakes has made about 0,1 dB. The additional code prize for Read-Solomon (255,41,215) code at correction on iteration of three additional mistakes has made about 0,17 dB.

Interpolation-Based Low-Complexity Chase Decoding Algorithms for Hermitian Codes

Algebraic-geometric (AG) codes have good error-correction capability due to their generally large code word length. However, their decoding remains complex, preventing practical applications. Addressing the challenge, this paper proposes two interpolation-based low-complexity Chase (LCC) decoding algorithms for one of the most popular AG codes-Hermitian codes. By choosing $\eta $ unreliable symbols and realizing them with the two most likely decisions, $2^\eta $ decoding test-vectors can be formulated. The first LCC algorithm performs interpolation for the common elements of the test-vectors, producing an intermediate outcome that will be shared by the uncommon element interpolation. It eliminates the redundant computation for decoding each test-vector, resulting in a low-complexity. With an interpolation multiplicity of one, the decoding is further facilitated by removing the requirement of pre-calculating the Hermitian curve’s corresponding coefficients. The second LCC algorithm is an adaptive variant of the first algorithm, where the number of test-vectors is determined by the reliability of received information. When the channel condition improves, it can reduce the complexity without compromising the decoding performance. Simulation results show that the both LCC algorithms outperform a number of existing algebraic decoding algorithms for Hermitian codes. Finally, our complexity analysis will reveal the proposals’ low-complexity feature.

Soft input decoder for high‐rate generalised concatenated codes

Generalised concatenated (GC) codes are well suited for error correction in flash memories for high-reliability data storage. The GC codes are constructed from inner extended binary Bose-Chaudhuri-Hocquenghem (BCH) codes and outer Reed-Solomon codes. The extended BCH codes enable high-rate GC codes and low-complexity soft input decoding. This work proposes a decoder architecture for high-rate GC codes. For such codes, outer error and erasure decoding are mandatory. A pipelined decoder architecture is proposed that achieves a high data throughput with hard input decoding. In addition, a low-complexity soft input decoder is proposed. This soft decoding approach combines a bit-flipping strategy with algebraic decoding. The decoder components for the hard input decoding can be utilised which reduces the overhead for the soft input decoding. Nevertheless, the soft input decoding achieves a significant coding gain compared with hard input decoding.

Algebraic Decoding of Cyclic Codes Using Partial Syndrome Matrices

Cyclic codes have been widely used in many applications of communication systems and data storage systems. This paper proposes a new procedure for decoding cyclic codes up to actual minimum distance. The decoding procedure consists of two steps: 1) computation of known syndromes and 2) computation of error positions and error values simultaneously. To do so, a matrix whose all entries are syndromes is called syndrome matrix . A matrix whose entries are either syndromes or the elements of a finite field is said to be partial syndrome matrix . In this paper, two novel methods are presented to determine error positions and error values simultaneously and directly. The first method uses a new partial syndrome matrix along with Gaussian elimination. The partial syndrome matrices for binary (respectively, ternary) cyclic codes of lengths from 69 to 99 (respectively, 16 to 37) are tabulated. For some cyclic codes, the partial syndrome matrices contain unknown syndromes; the second method constructs a matrix from a system of equations, which is generated by the determinants of different partial syndrome matrices and makes use of Gaussian elimination to determine its row rank. Many more cyclic codes beyond the Bose–Chaudhuri–Hocquenghem bound can be decoded with these methods.

Spatially Coupled Split-Component Codes With Iterative Algebraic Decoding

We analyze a class of high performance, low decoding-data-flow error-correcting codes suitable for high bit-rate optical-fiber communication systems. A spatially coupled split-component ensemble is defined, generalizing from the most important codes of this class, staircase codes and braided block codes, and preserving a deterministic partitioning of component-code bits over code blocks. Our analysis focuses on low-complexity iterative algebraic decoding, which, for the binary erasure channel, is equivalent to a generalization of the peeling decoder. Using the differential equation method, we derive a vector recursion that tracks the expected residual graph evolution throughout the decoding process. The threshold of the recursion, for asymptotically long component codes, is found using potential function analysis. We generalize the analysis to mixture ensembles consisting of more than one type of component code. We give an example of a mixture ensemble consisting of two component codes, which has better performance than spatially-coupled split-component ensembles consisting of only one component code. The analysis extends to the binary symmetric channel by assuming miscorrection-free component-code decoding. Simple upper bounds on the number of errors correctable by the ensemble are derived. Finally, we analyze the threshold of spatially coupled split-component ensembles under beyond bounded-distance component decoding.

Computing Sparse GCD of Multivariate Polynomials via Polynomial Interpolation

The problem of computing the greatest common divisor (GCD) of multivariate polynomials, as one of the most important tasks of computer algebra and symbolic computation in more general scope, has been studied extensively since the beginning of the interdisciplinary of mathematics with computer science. For many real applications such as digital image restoration and enhancement, robust control theory of nonlinear systems, L1-norm convex optimization in compressed sensing techniques, as well as algebraic decoding of Reed-Solomon and BCH codes, the concept of sparse GCD plays a core role where only the greatest common divisors with much fewer terms than the original polynomials are of interest due to the nature of problems or data structures. This paper presents two methods via multivariate polynomial interpolation which are based on the variation of Zippel’s method and Ben-Or/Tiwari algorithm, respectively. To reduce computational complexity, probabilistic techniques and randomization are employed to deal with univariate GCD computation and univariate polynomial interpolation. The authors demonstrate the practical performance of our algorithms on a significant body of examples. The implemented experiment illustrates that our algorithms are efficient for a quite wide range of input.

An analogue of the ℤ4-Goethals code in non-primitive length

This paper constructs a cyclic ℤ4-code with a parity-check matrix similar to that of Goethals code but in length 2m + 1, for all m ≥ 4. This code is a subcode of the lifted Zetterberg code for m even. Its minimum Lee weight is shown to be at least 10, in general, and exactly 12 in lengths 33, 65. The authors give an algebraic decoding algorithm which corrects five errors in these lengths for m = 5, 6 and four errors for m > 6.

Edge Coloring and Stopping Sets Analysis in Product Codes With MDS Components

We consider non-binary product codes with MDS components and their iterative row-column algebraic decoding on the erasure channel. Both independent and block erasures are considered in this paper. A compact graph representation is introduced on which we define double-diversity edge colorings via the rootcheck concept. An upper bound of the number of decoding iterations is given as a function of the graph size and the color palette size $M$ . Then, we propose a differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols. The complexity of this algorithm per iteration is $o(M^{\aleph })$ , for a given differential evolution parameter $\aleph $ , where $M^{\aleph }$ itself is small with respect to the huge cardinality of the coloring ensemble. Stopping sets of a product code are defined in the context of MDS components and a relationship is established with the graph representation. A full characterization of these stopping sets is given up to a size $(d+1)^{2}$ , where $d$ is the minimum Hamming distance of the MDS component code. The performance of MDS-based product codes with and without double-diversity coloring is analyzed in presence of both the block and the independent erasures. In the latter case, ML and iterative decoding are proven to coincide at small channel erasure probability. Furthermore, numerical results show excellent performance in presence of unequal erasure probability due to double-diversity colorings.

Pade approximants and key lattices for decoding alternant codes

In this paper we present two methods for algebraic decoding of alternant codes. The paper recalls fundamental facts of these codes and the concept of key equation introduced by Berlekamp on which decoding is based. The first method is based on Pade approximants. The second method consist of translating the key equation into a lattice over the ring of polynomials, called key lattice which depends on the syndrome, and then seek a shortest vector in this key lattice, with degrees constraints. The key lattice approach, following the fruitfull formulation of H. W. Lenstra, is based on a result of Grigor'ev showing that computing a minimal vector in the mentioned lattice is polynomial. The computational complexity of the proposed methods are given in terms of the error-correction capability of the codes and results in tractable polynomial time algorithms. A fast parallel version of the key lattice method is deduced.

Progressive algebraic Chase decoding algorithms for Reed–Solomon codes

This study proposes a progressive algebraic Chase decoding (PACD) algorithm for Reed–Solomon (RS) codes. On the basis of the received information, 2 η (η > 0) interpolation test-vectors are constructed for the interpolation-based algebraic Chase decoding. A test-vector reliability function is defined to assess their potential for yielding the intended message. The algebraic Chase decoding will then be performed progressively granting priority to decode the test-vectors that are more likely to yield the message, and is then terminated once it is found. Consequently, the decoding complexity can be adapted to the quality of the received information. An enhanced-PACD (E-PACD) algorithm is further proposed by coupling the PACD algorithm with the adaptive belief propagation (ABP) decoding. The ABP decoding generates new test-vectors for the PACD algorithm by enhancing the received information. It improves the Chase decoding performance without increasing the decoding complexity exponentially. It is shown that the E-PACD algorithm's complexity can be significantly reduced by utilising the existing interpolation information of the previous Chase decodings'. Our performance evaluations show that the two proposed decoders outperform a number of existing algebraic decoding approaches. Complexity and memory analyses of the PACD algorithm are also presented, demonstrating that this is an efficient RS decoding strategy.

Algebraic Decoding of Cyclic Codes Without Error-Locator Polynomials

The algebraic decoding of a $p$ -ary cyclic code consists of four steps: 1) computation of the known syndromes using the received word; 2) computation of the unknown syndromes from the known syndromes; 3) computation of the error positions by a use of the Berlekamp-Massey (BM) algorithm and Chien’s search; and 4) computation of the error values by solving a linear system. This paper addresses two problems of determining the error positions and computing the unknown syndromes. To solve the first problem, a new matrix, together with Gaussian elimination instead of the BM algorithm and Chien’s search, is proposed. In this new simplified decoder, finding an error-locator polynomial is completely avoided. A main advantage of the presented decoding method is when the Bose-Chaudhuri–Hocquenghem bound is unequal to the minimum distance of the code. Some cyclic codes, which do not have $2t$ consecutive known syndromes, can be decoded up to their actual minimum distance by using the presented matrix once. To solve the second problem, two algorithms for different square matrices reported recently by Lee et al. are also provided to calculate the value of an unknown syndrome. Finally, an algebraic decoding of the $(47,23,15)$ ternary cyclic code is given.

Algebraic decoding of the (71, 36, 11) quadratic residue code

In this study, a new approach is developed to facilitate faster decoding of a binary systematic (71, 36, 11) quadratic residue (QR) code. In this decoder, it simplifies the step of calculating the condition and avoids calculating the unknown syndrome, thereby yielding a fast algebraic decoder for correcting four possible errors. Moreover, while using the proposed algorithm, if uses the channel measurement information proposed by Chase to sequentially invert the bits of the received word until one of the errors is cancelled for the five-error case and apply the new algebraic decoding algorithm mentioned above to correct the remaining four errors, the algorithm has been verified through a software simulation in C-language. The simulation shows that the decoding scheme developed here is more efficient than the previous decoding algorithm developed for the (71, 36, 11) QR code and it is naturally suitable for software implementation.

Algebraic decoding of folded Gabidulin codes

An efficient interpolation-based decoding algorithm for $$h$$h-folded Gabidulin codes is presented that can correct rank errors beyond half the minimum rank distance for any code rate $$0\le R\le 1$$0≤R≤1. The algorithm serves as a list decoder or as a probabilistic unique decoder and improves upon existing schemes, especially for high code rates. A probabilistic unique decoder with adjustable decoding radius is presented. The decoder outputs a unique solution with high probability and requires at most $$\mathcal {O}({s^2n^2})$$O(s2n2) operations in $$\mathbb {F}_{q^m}$$Fqm, where $$1\le s\le h$$1≤s≤h is a decoding parameter and $$n\le m$$n≤m is the length of the unfolded code over $$\mathbb {F}_{q^m}$$Fqm. An upper bound on the average list size of folded Gabidulin codes and on the decoding failure probability of the decoder is given. Applying the ideas to a list decoding algorithm by Mahdavifar and Vardy (List-decoding of subspace codes and rank-metric codes up to Singleton bound, ISIT 2012) improves the performance when used as probabilistic unique decoder and gives an upper bound on the failure probability.

Algebraic Decoding of BCH Codes over Symbol-Pair Read Channels: Cases of Two-Pair and Three-Pair Error Correction

Algebraic Decoding of BCH Codes over Symbol-Pair Read Channels: Cases of Two-Pair and Three-Pair Error Correction

Spectral Analysis of Quasi-Cyclic Product Codes

This paper considers a linear quasi-cyclic product code of two given quasi-cyclic codes of relatively prime lengths over finite fields. We give the spectral analysis of a quasi-cyclic product code in terms of the spectral analysis of the row and column codes. Moreover, we provide a new lower bound on the minimum Hamming distance of a given quasi-cyclic code and present a new algebraic decoding algorithm. More specifically, we prove an explicit (unreduced) basis of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code in terms of the generator matrix in reduced Grobner basis with respect to the position-over-term (RGB/POT) order form of the $ {\ell _{A}}$ -quasi-cyclic row code and the $ {\ell _{B}}$ -quasi-cyclic column code, respectively. This generalizes the work of Burton and Weldon for the generator polynomial of a cyclic product code (where $ {\ell _{A}}= {\ell _{B}}=1$ ). Furthermore, we derive the generator matrix in Pre-RGB/POT form of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code for two special cases: i) for $ {\ell _{A}}=2$ and $ {\ell _{B}}=1$ and ii) if the row code is a one-level $ {\ell _{A}}$ -quasi-cyclic code (for arbitrary $ {\ell _{A}}$ ) and $ {\ell _{B}}=1$ . For arbitrary $ {\ell _{A}}$ and $ {\ell _{B}}$ , the Pre-RGB/POT form of the generator matrix of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code is conjectured. The spectral analysis is applied to the generator matrix of the product of an $ {\ell }$ -quasi-cyclic and a cyclic code, and we propose a new lower bound on the minimum Hamming distance of a given $ {\ell }$ -quasi-cyclic code. In addition, we develop an efficient syndrome-based decoding algorithm for $ {\ell }$ -phased burst errors with guaranteed decoding radius.

On the lifted Zetterberg code

The even-weight subcode of a binary Zetterberg code is a cyclic code with generator polynomial $$g(x)=(x+1)p(x)$$g(x)=(x+1)p(x), where p(x) is the minimum polynomial over GF(2) of an element of order $$2^m+1$$2m+1 in $$GF(2^{2m})$$GF(22m) and m is even. This even binary code has parameters $$[2^m+1,2^m-2m, 6]$$[2m+1,2m-2m,6]. The quaternary code obtained by lifting the code to the alphabet $${\mathbb {Z}}_4=\{0,1,2,3\}$$Z4={0,1,2,3} is shown to have parameters $$[2^m+1,2^m-2m, d_L ]$$[2m+1,2m-2m,dL], where $$d_L \ge 8$$dL?8 denotes the minimum Lee distance. The image of the Gray map of the lifted code is a binary code with parameters $$(2^{m+1}+2,2^k,d_H)$$(2m+1+2,2k,dH), where $$d_H \ge 8$$dH?8 denotes the minimum Hamming weight and $$k=2^{m+1}-4m$$k=2m+1-4m. For $$m=6$$m=6 these parameters equal the parameters of the best known binary linear code for this length and dimension. Furthermore, a simple algebraic decoding algorithm is presented for these $${\mathbb {Z}}_4$$Z4-codes for all even m. This appears to be the first infinite family of $${\mathbb {Z}}_4$$Z4-codes of length $$n=2^m+1$$n=2m+1 with $$d_L \ge 8$$dL?8 having an algebraic decoding algorithm.

On the lifted Melas code

The binary Melas code is a cyclic code with generator polynomial g(u)=p(u)p(u)? where p(u) is a primitive polynomial of odd degree m?5 and the ? denotes reciprocation. The even-weight subcode of a Melas code has generator polynomial (u+1)g(u) and parameters [2m?1,2m?2m?2,6]. This code is lifted to ?4$\mathbb {Z}_{4}$ and the quaternary code is shown to have parameters [2m?1,2m?2m?2,dL?8], where dL denotes the minimum Lee distance. An algebraic decoding algorithm correcting all errors of Lee weight ≤3 is presented for this code. The Gray map of this quaternary code is a binary code with parameters [2m+1?2,2m+1?4m?4,dH?8] where dH is the minimum Hamming distance. For m=5,7 the minimum distance equals the minimum distance of the best known linear code for the given length and code size.

Algebraic Decoding of Some Quadratic Residue Codes With Weak Locators

In this paper, an explicit expression of the weak-locator polynomial for p-ary quadratic residue codes is presented by a modification of the Feng-Tzeng matrix method. The differences between the modified version and the original Feng-Tzeng matrix are that in the new matrix, not every entry is a syndrome, and every syndrome entry is a known syndrome. By utilizing this technique, an algebraic decoding of the ternary (61, 30, 12) quadratic residue code is proposed. This new result has never been seen in the literature to our knowledge. An advantage of the proposed decoding algorithm is that in general the obtained weak-locator polynomials can decode efficiently not only all the error patterns of weights four and five, but also some error patterns of weight six.