Projection Decoding of Some Binary Optimal Linear Codes of Lengths 36 and 40

Constacyclic codes over finite fields are a family of linear codes and contain cyclic codes as a subclass. Constacyclic codes are closely related to many areas of mathematics and outperform cyclic codes in several aspects. Hence, constacyclic codes are of theoretical importance. On the other hand, constacyclic codes are important in practice, as they have rich algebraic structures and may have efficient decoding algorithms. In this extended abstract, two classes of constacyclic codes are constructed using a general construction of constacyclic codes with cyclic codes. The first class of constacyclic codes is motivated by the punctured Dilix cyclic codes, and the second class is motivated by the punctured generalised Reed-Muller codes. The two classes of constacyclic codes contain optimal linear codes. The parameters of the two classes of constacyclic codes are analysed, and some open problems are presented in this extended abstract.

Error-correcting codes are combinatorial objects designed to cope with the problem of reliable transmission of information on a noisy channel. A fundamental algorithmic challenge in coding theory and practice is to efficiently decode the original transmitted message even when a few symbols of the received word are in error. The naive search algorithm runs in exponential time, and several classical polynomial time decoding algorithms are known for specific code families. Traditionally, however, these algorithms have been constrained to output a unique codeword. Thus they faced a “combinatorial barrier” and could only correct up to d/2 errors, where d is the minimum distance of the code. An alternate notion of decoding called list decoding, proposed independently by Elias and Wozencraft in the late 50s, allows the decoder to output a list of all codewords that differ from the received word in a certain number of positions. Even when constrained to output a relatively small number of answers, list decoding permits recovery from errors well beyond the d/2 barrier, and opens up the possibility of meaningful error-correction from large amounts of noise. However, for nearly four decades after its conception, this potential: of list decoding was largely untapped due to the lack of efficient algorithms to list decode beyond d/2 errors for useful families of codes. This thesis presents a detailed investigation of list decoding, and proves its potential, feasibility, and importance as a combinatorial and algorithmic concept. We prove several; combinatorial results that sharpen our understanding of the potential and limits of list; decoding, and its relation to more classical parameters like the rate and minimum distance. The crux of the thesis is its algorithmic results, which were lacking in the early works on list decoding. Our algorithmic results include: (1) Efficient list decoding algorithms for classically studied codes such as Reed-Solomon codes and algebraic-geometric codes. In particular, building upon an earlier algorithm due to Sudan, we present the first polynomial time algorithm to decode Reed-Solomon codes beyond d/2 errors for every value of the rate. (2) A new soft list decoding algorithm for Reed-Solomon and algebraic-geometric codes and novel decoding algorithms for concatenated codes based on it. (3) New code constructions using concatenation and/or expander graphs that have good (and sometimes near-optimal) rate and are efficiently list decodable from extremely large amounts of noise. (4) Expander-based constructions of linear time encodable and decodable codes that ca4 correct up to the maximum possible fraction of errors, using unique (not list) decoding. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)

Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, a new Gray map between codes over Fp + uFp + u2Fp and codes over Fp is defined, where p is an odd prime. By means of this map, it is shown that the Gray image of a linear (1+u+u2)-constacyclic code over Fp + uFp + u2Fp of length n is a repeated-root cyclic code over Fp of length pn. Furthermore, some examples of optimal linear cyclic codes over F3 from (1 + u + u2)-constacyclic codes over F3 + uF3 + u2F3 are given.

We consider locally repairable codes over small fields and propose constructions of optimal cyclic and linear codes in terms of the dimension for a given distance and length. Four new constructions of optimal linear codes over small fields with locality properties are developed. The first two approaches give binary cyclic codes with locality two. While the first construction has availability one, the second binary code is characterized by multiple available repair sets based on a binary Simplex code. The third approach extends the first one to q-ary cyclic codes including (binary) extension fields, where the locality property is determined by the properties of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear codes with locality greater than two are obtained by the fourth construction.

The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm.

Preface. Foreword. The ECC web site. 1. Introduction. 1.1 Error correcting coding: Basic concepts. 1.1.1 Block codes and convolutional codes. 1.1.2 Hamming distance, Hamming spheres and error correcting capability. 1.2 Linear block codes. 1.2.1 Generator and parity-check matrices. 1.2.2 The weight is the distance. 1.3 Encoding and decoding of linear block codes. 1.3.1 Encoding with G and H. 1.3.2 Standard array decoding. 1.3.3 Hamming spheres, decoding regions and the standard array. 1.4 Weight distribution and error performance. 1.4.1 Weight distribution and undetected error probability over a BSC. 1.4.2 Performance bounds over BSC, AWGN and fading channels. 1.5 General structure of a hard-decision decoder of linear codes. Problems. 2. Hamming, Golay and Reed-Muller codes. 2.1 Hamming codes. 2.1.1 Encoding and decoding procedures. 2.2 The binary Golay code. 2.2.1 Encoding. 2.2.2 Decoding. 2.2.3 Arithmetic decoding of the extended (24, 12, 8) Golay code. 2.3 Binary Reed-Muller codes. 2.3.1 Boolean polynomials and RM codes. 2.3.2 Finite geometries and majority-logic decoding. Problems. 3. Binary cyclic codes and BCH codes. 3.1 Binary cyclic codes. 3.1.1 Generator and parity-check polynomials. 3.1.2 The generator polynomial. 3.1.3 Encoding and decoding of binary cyclic codes. 3.1.4 The parity-check polynomial. 3.1.5 Shortened cyclic codes and CRC codes. 3.1.6 Fire codes. 3.2 General decoding of cyclic codes. 3.2.1 GF(2m) arithmetic. 3.3 Binary BCH codes. 3.3.1 BCH bound. 3.4 Polynomial codes. 3.5 Decoding of binary BCH codes. 3.5.1 General decoding algorithm for BCH codes. 3.5.2 The Berlekamp-Massey algorithm (BMA). 3.5.3 PGZ decoder. 3.5.4 Euclidean algorithm. 3.5.5 Chien search and error correction. 3.5.6 Errors-and-erasures decoding. 3.6 Weight distribution and performance bounds. 3.6.1 Error performance evaluation. Problems. 4. Nonbinary BCH codes: Reed-Solomon codes. 4.1 RS codes as polynomial codes. 4.2 From binary BCH to RS codes. 4.3 Decoding RS codes. 4.3.1 Remarks on decoding algorithms. 4.3.2 Errors-and-erasures decoding. 4.4 Weight distribution. Problems. 5. Binary convolutional codes. 5.1 Basic structure. 5.1.1 Recursive systematic convolutional codes. 5.1.2 Free distance. 5.2 Connections with block codes. 5.2.1 Zero-tail construction. 5.2.2 Direct-truncation construction. 5.2.3 Tail-biting construction. 5.2.4 Weight distributions. 5.3 Weight enumeration. 5.4 Performance bounds. 5.5 Decoding: Viterbi algorithm with Hamming metrics. 5.5.1 Maximum-likelihood decoding and metrics. 5.5.2 The Viterbi algorithm. 5.5.3 Implementation issues. 5.6 Punctured convolutional codes. 5.6.1 Implementation issues related to punctured convolutional codes. 5.6.2 RCPC codes. Problems. 6. Modifying and combining codes. 6.1 Modifying codes. 6.1.1 Shortening. 6.1.2 Extending. 6.1.3 Puncturing. 6.1.4 Augmenting, expurgating and lengthening. 6.2 Combining codes. 6.2.1 Time sharing of codes. 6.2.2 Direct sums of codes. 6.2.3 The |u|u + v|-construction and related techniques. 6.2.4 Products of codes. 6.2.5 Concatenated codes. 6.2.6 Generalized concatenated codes. 7. Soft-decision decoding. 7.1 Binary transmission over AWGN channels. 7.2 Viterbi algorithm with Euclidean metric. 7.3 Decoding binary linear block codes with a trellis. 7.4 The Chase algorithm. 7.5 Ordered statistics decoding. 7.6 Generalized minimum distance decoding. 7.6.1 Sufficient conditions for optimality. 7.7 List decoding. 7.8 Soft-output algorithms. 7.8.1 Soft-output Viterbi algorithm. 7.8.2 Maximum-a posteriori (MAP) algorithm. 7.8.3 Log-MAP algorithm. 7.8.4 Max-Log-MAP algorithm. 7.8.5 Soft-output OSD algorithm. Problems. 8. Iteratively decodable codes. 8.1 Iterative decoding. 8.2 Product codes. 8.2.1 Parallel concatenation: Turbo codes. 8.2.2 Serial concatenation. 8.2.3 Block product codes. 8.3 Low-density parity-check codes. 8.3.1 Tanner graphs. 8.3.2 Iterative hard-decision decoding: The bit-flip algorithm. 8.3.3 Iterative probabilistic decoding: Belief propagation. Problems. 9. Combining codes and digital modulation. 9.1 Motivation. 9.1.1 Examples of signal sets. 9.1.2 Coded modulation. 9.1.3 Distance considerations. 9.2 Trellis-coded modulation (TCM). 9.2.1 Set partitioning and trellis mapping. 9.2.2 Maximum-likelihood. 9.2.3 Distance considerations and error performance. 9.2.4 Pragmatic TCM and two-stage decoding. 9.3 Multilevel coded modulation. 9.3.1 Constructions and multistage decoding. 9.3.2 Unequal error protection with MCM. 9.4 Bit-interleaved coded modulation. 9.4.1 Gray mapping. 9.4.2 Metric generation: De-mapping. 9.4.3 Interleaving. 9.5 Turbo trellis-coded modulation. 9.5.1 Pragmatic turbo TCM. 9.5.2 Turbo TCM with symbol interleaving. 9.5.3 Turbo TCM with bit interleaving. Problems. Appendix A: Weight distributions of extended BCH codes. A.1 Length 8. A.2 Length 16. A.3 Length 32. A.4 Length 64. A.5 Length 128. Bibliography. Index.

A class of optimal three-weight [q^k-1,k+1,q^{k-1}(q-1)-1] cyclic codes over {mathrm{I!F}}_q, with kge 2, achieving the Griesmer bound, was presented by Heng and Yue (IEEE Trans Inf Theory 62(8):4501–4513, 2016. https://doi.org/10.1109/TIT.2016.2550029). In this paper we study some of the subfield codes of this class of optimal cyclic codes when k=2. The weight distributions of the subfield codes are settled. It turns out that some of these codes are optimal and others have the best known parameters. The duals of the subfield codes are also investigated and found to be almost optimal with respect to the sphere-packing bound. In addition, the covering structure for the studied subfield codes is determined. Some of these codes are found to have the important property that any nonzero codeword is minimal, which is a desirable property that is useful in the design of a secret sharing scheme based on a linear code. Moreover, a specific example of a secret sharing scheme based on one of these subfield codes is given. Finally, a class of optimal two-weight linear codes over {mathrm{I!F}}_q, achieving the Griesmer bound, whose duals are almost optimal with respect to the sphere-packing bound is presented. Through a different approach, this class of optimal two-weight linear codes was reported very recently by Heng (IEEE Trans Inf Theory 69(2):978–994, 2023. https://doi.org/10.1109/TIT.2022.3203380). Furthermore, it is shown that these optimal codes can be used to construct strongly regular graphs.

Time synchronization is crucial for the safe and reliable operation of the fifth generation (5G) network, especially for applications requiring ultra-reliable low-latency data transmissions. The time synchronization problem, however, becomes increasingly challenging in high-mobility scenarios because the channel conditions, e.g., the multipath delay spread may vary rapidly. While there exist numerous works on the design of efficient channel decoding algorithms, decoding linear codes such as polar codes in the presence of synchronization errors is a less-explored topic. In this paper, we aim to fill this void and develop a systemic approach to decode general binary linear codes over binary symmetric channels with synchronization errors in which the lack of synchronization is modeled as the deletion channel model. The maximum likelihood (ML) decoding problem for binary linear codes over deletion channels is first formulated as a nonlinear optimization problem, in which a set of linear constraints are employed to characterize the input-output relationship of a deletion channel. It turns out that both the objective function and the constraints of this optimization problem are nonlinear, which poses significant challenges against the design of efficient decoding algorithms. As a remedy, we first replace the nonlinear objective function of this optimization problem via a lower bound. And we prove this lower bound is a linear function in the special case that the input is binary. We then apply the linear programming (LP) relaxation approach to obtain an approximate solution to the proposed nonlinear optimization problem. An adaptive branch-and-cut decoding algorithm has also been developed by making use of the ML-certificate property of the LP decoder for deletion channel. It is seen through simulation studies that the proposed decoding algorithm can achieve close-to-optimal bit error rate (BER) decoding performance at moderate computational complexity.

New ternary codes of dimension 6 are presented which improve the bounds on optimal linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a greedy algorithm. This work extends previous results on QT codes of dimension 6. In particular, several new two-weight QT codes are presented. Numerous new optimal codes which meet the Griesmer bound are given, as well as others which establish lower bounds on the maximum minimum distance.

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.

Using methods originating in numerical analysis, we will develop a unified framework for derivation of efficient list decoding algorithms for algebraicgeometric codes. We will demonstrate our method by accelerating Sudan's list decoding algorithm for Reed-Solomon codes [22], its generalization to algebraic-geometric codes by Shokrollahi and Wasserman [21], and the recent improvement of Guruswami and Sudan [8] in the case of ReedSolomon codes. The basic problem we attack in this paper is that of efficiently finding nonzero elements in the kernel of a structured matrix. The structure of such an n x n-matrix allows it to be to ? n parameters for some ? which is usually a constant in applications. The concept of structure is formalized using the displacement operator. The displacement operator allows to perform matrix operations on the compressed version of the matrix. In particular, we can find a PLU- decomposition of the original matrix in time O(? n2), which is quadratic in n for constant ?. We will derive appropriate displacement operators for matrices that occur in the context of list decoding, and apply our general algorithm to them. For example, we will obtain algorithms that use O(n2 l) and O(n7/3 l) operations over the base field for list decoding of Reed-Solomon codes and algebraic-geometric codes from certain plane curves, respectively, where l is the length of the list. Assuming that l is constant, this gives algorithms of running time O(n2) and O(n7/3), which is the same as the running time of conventional decoding algorithms. We will also sketch methods to parallelize our algorithms

The performance and the decoding complexity of a novel coding scheme based on the concatenation of maximum distance separable (MDS) codes and linear random fountain codes are investigated. Differently from Raptor codes (which are based on a serial concatenation of a high-rate outer block code and an inner Luby-transform code), the proposed coding scheme can be seen as a parallel concatenation of a MDS code and a linear random fountain code, both operating on the same finite field. Upper and lower bounds on the decoding failure probability under maximum-likelihood (ML) decoding are developed. It is shown how, for example, the concatenation of a (15,10) Reed-Solomon (RS) code and a linear random fountain code over a finite field of order 16, {F}_{16}, brings to a decoding failure probability 4 orders of magnitude lower than the one of a linear random fountain code for the same receiver overhead in a channel with a erasure probability of ε=5\cdot10^{-2}. It is illustrated how the performance of the novel scheme approaches that of an idealized fountain code for higher-order fields and moderate erasure probabilities. An efficient decoding algorithm is developed for the case of a (generalized) RS code.

The performance and the decoding complexity of a novel coding scheme based on the concatenation of maximum distance separable (MDS) codes and linear random fountain codes are investigated. Differently from Raptor codes (which are based on a serial concatenation of a high-rate outer block code and an inner Luby-transform code), the proposed coding scheme can be seen as a parallel concatenation of a MDS code and a linear random fountain code, both operating on the same finite field. Upper and lower bounds on the decoding failure probability under maximum-likelihood (ML) decoding are developed. It is shown how, for example, the concatenation of a $(15,10)$ Reed-Solomon (RS) code and a linear random fountain code over a finite field of order $16$, $\mathbb {F}_{16}$, brings to a decoding failure probability $4$ orders of magnitude lower than the one of a linear random fountain code for the same receiver overhead in a channel with a erasure probability of $ε=5\cdot10^{-2}$. It is illustrated how the performance of the novel scheme approaches that of an idealized fountain code for higher-order fields and moderate erasure probabilities. An efficient decoding algorithm is developed for the case of a (generalized) RS code.

There has been renewed interest in iterative decod- ing algorithms for concatenated codes since the introduction of Turbo codes. Such iterative methods are built on top of soft-in- soft-out algorithms. Powerful concatenated codes may be con- structed with linear block codes codes as constituent codes. Such codes can be better alternatives to Turbo Codes either when high coding rates or when short interleaver lengths are required. This paper presents a computationally efficient maximum a-posteriori (MAP) soft-in-soft-out (SISO) algorithm for RS codes and related codes. Index Terms—Reed-Solomon codes, soft decoding, iterative de- coding, concatenated codes, SISO algorithms. I. I NTRODUCTION binary images of RS codes can be represented as linear combi- nations of two sub-field sub-codes which results in an efficient decoding algorithm. We exploit this property to derive an effi- cient soft-output MAP decoding algorithm for RS codes. The algorithm can also be applied to some sub-codes of RS codes, which as we will show later have good performance. The al- gorithm is also well suited for hardware implementation, as a significant number of computations may be performed in par- allel. We also show that the proposed algorithm is many orders lower in complexity compared to MAP algorithm applied on the Wolf trellis. The remainder of this paper is organized as follows. Section II gives a brief overview of algebraic properties of RS codes. The proposed decoding algorithm is presented in Section III. Simulation results for the proposed algorithm applied on prod- uct RS codes are given in Section IV. Finally, Section V, gives conclusions and directions for future work.

The McEliece cryptosystem is one of the oldest public-key cryptosystems ever designed. It is also the first public-key cryptosystem based on linear error-correcting codes. Its main advantage is to have very fast encryption and decryption functions. However it suffers from a major drawback. It requires a very large public key which makes it very difficult to use in many practical situations. A possible solution is to advantageously use quasi-cyclic codes because of their compact representation. On the other hand, for a fixed level of security, the use of optimal codes like Maximum Distance Separable ones allows to use smaller codes. The almost only known family of MDS codes with an efficient decoding algorithm is the class of Generalized Reed-Solomon (GRS) codes. However, it is well-known that GRS codes and quasi-cyclic codes do not represent secure solutions. In this paper we propose a new general method to reduce the public key size by constructing quasi-cyclic Alternant codes over a relatively small field like \({\mathbb{F}}_{2^8}\). We introduce a new method of hiding the structure of a quasi-cyclic GRS code. The idea is to start from a Reed-Solomon code in quasi-cyclic form defined over a large field. We then apply three transformations that preserve the quasi-cyclic feature. First, we randomly block shorten the RS code. Next, we transform it to get a Generalised Reed Solomon, and lastly we take the subfield subcode over a smaller field. We show that all existing structural attacks are infeasible. We also introduce a new NP-complete decision problem called quasi-cyclic syndrome decoding. This result suggests that decoding attack against our variant has little chance to be better than the general one against the classical McEliece cryptosystem. We propose a system with several sizes of parameters from 6,800 to 20,000 bits with a security ranging from 280 to 2120.Keywordspublic-key cryptographyMcEliece cryptosystemAlternant codequasi-cyclic