A decoding method of an n length binary BCH code through (n + 1)n length binary cyclic code

Deep holes are very important in the decoding of generalized RS codes, and deep holes of RS codes have been widely studied, but there are few works on constructing general linear codes based on deep holes. Therefore, we consider constructing binary linear codes by combining deep holes with binary BCH codes. In this article, we consider the 2-error-correcting binary primitive BCH codes and the extended codes to construct new binary linear codes by combining them with deep holes, respectively. Furthermore, three classes of binary linear codes are constructed, and then we determine the parameters and the weight distributions of these new binary linear codes.

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The known construction of cyclic concatenated codes is based on the fact that the inner is a cyclic minimal code, the outer code is cyclic, and the lengths of the inner and outer codes are relatively prime. It is shown that if the outer code is a suitably chosen constacyclic code the overall concatenated code is always cyclic regardless of the length of the outer code. Moreover, it follows that any cyclic code of composite length is a direct sum of cyclic concatenated codes with inner cyclic minimal codes and outer constacyclic codes. This description of cyclic codes of composite length leads to the introduction of the concept of a poor-code length (PCL). All but low-rate codes of this length have a poor minimum distance. A PCL is directly related to the existence of irreducible binomials. In the binary case, the first few PCLs are 9, 25, 27, 45, 49, 75, 81 and 99. Arbitrarily long binary cyclic codes that are better than binary BCH codes of primitive length are constructed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The conventional parity-check matrix transformation algorithm (PTA) requires matrix inversion to transform matrices of maximum distance separable (MDS) codes. However, such matrix transformation is not always guaranteed for the class of non-MDS codes. Hence, the PTA fails for binary cyclic (BC) codes. To overcome this limitation, we developed a generalized parity-check matrix transformation (GPT) algorithm for binary cyclic codes. The GPT avoids the matrix inversion step of the PTA. It permutes the columns of the parity-check matrix based on the reliability information from the channel. Results show a significant bit error rate (BER) performance gain as compared to the existing PTA. It also presents a reasonable BER performance as compared to the other soft-decision (SD) decoding algorithms. In addition, the decoder functions within a practical decoding time complexity, particularly at the message passing stage.

An [$$n, k, d$$n,k,d] code is a binary linear code of block length $$n$$n, dimension $$k$$k and minimum Hamming distance $$d$$d. Since the minimum distance determines the error detection or correction capability, it is desired that $$d$$d is as large as possible for the given block length $$n$$n and dimension $$k$$k. One of the most fundamental problems in coding theory is to construct codes with best possible minimum distances. This problem is very difficult in both theory and practice. During the last decades, it has proved that the class of quasi-cyclic (QC) codes contain many such codes. In this paper, augmentation of binary QC codes is studied. A new augmentation algorithm is presented, and 10 new $$h$$h-generator QC codes that are better than previously known code have been constructed. Furthermore, Construction X has been applied to obtain another 18 new improved binary linear codes. With the standard construction techniques, a total of 124 new binary linear codes that improve the lower bound on the minimum distance have been obtained.

By introducing the ideas of reducing computation complexity in the fast Fourier transform (FFT) to decoding, a low complexity decoding of binary BCH codes is derived which transforms the determination of one-dimensional error word into two-dimensional matrix. For the case of double-error-correcting binary BCH codes, the frequency domain decoding based on prime-factor FFT (in short, PF-FDD) was designed. Compared with the conventional frequency domain decoding algorithm, PF-FDD has the advantages of lower decoding complexity and faster decoding speed at the expense of increasing storage memory. When compared with the ordinary look-up table decoding scheme, the error patterns to store are decreased greatly. Hence, PF-FDD achieves a better tradeoff between decoding complexity and hardware complexity. The proposed low complexity decoding scheme is suitable for binary BCH codes whose block length has prime factors and error correcting capacity is relatively small, and is expected to apply to soft-decision decoding.

This work is the further development of the theory of norms of syndromes: the theory of polynomial invariants of G-orbits of errors expands with the group G of automorphisms of binary cyclic BCH codes obtained by joining the degrees of cyclotomic permutation to the group Γ and practically exhausting the group of automorphisms of BCH codes. It is determined that polynomial invariants, like the norms of syndromes, have a scalar character and are one-to-one characteristics of their orbits for BCH codes with a constructive distance of five. The paper introduces the corresponding vector polynomial invariants for primitive cyclic BCH codes with a constructive distance of seven, next to the norms of the syndromes that are already vector quantities; the basic properties of the vector polynomial invariants are investigated. It is established that the property of mutual unambiguity is violated: there are G-orbit-isomers, which are different, but have the same vector polynomial invariants. It is substantiated and demonstrated by examples that this circumstance greatly complicates error decoding algorithms based on polynomial invariants

A new upper bound on the minimum distance of binary cyclic arithmetic codes of composite length is derived. New classes of binary cyclic arithmetic codes of composite length are introduced. The error correction capability of these codes is discussed, and in some cases the actual minimum distance is found. Decoding algorithms based on majority-logic decision are proposed for these codes.

This paper considers the optimal generator matrices of a given binary cyclic code over a binary symmetric channel with crossover probability p → 0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. Our goal here is to seek the optimal realization of encoding functions by taking advantage of the structure of the codes, and to derive the probability of information bit error when possible. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many cyclic codes with such properties. We also present some binary cyclic codes whose optimal generator matrices are non-systematic under complete decoding.

This paper considers the optimal generator matrices of a binary cyclic code over a binary symmetric channel with crossover probability p rarr 0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many binary cyclic codes satisfying these properties.

The problem of efficient maximum-likelihood soft decision decoding of binary BCH codes is considered. It is known that those primitive BCH codes whose designed distance is one less than a power of two, contain subcodes of high dimension which consist of a direct-sum of several identical codes. The authors show that the same kind of direct-sum structure exists in all the primitive BCH codes, as well as in the BCH codes of composite block length. They also introduce a related structure termed the concurring-sum, and then establish its existence in the primitive binary BCH codes. Both structures are employed to upper bound the number of states in the minimal trellis of BCH codes, and develop efficient algorithms for maximum-likelihood soft decision decoding of these codes. >

It has been well-known that the class of quasi-cyclic (QC) codes contain many good codes. In thispaper, a method to conduct a computer search for binary 2-generator QC codes is presented, anda large number of good 2-generator QC codes have been obtained. 5 new binary QC codes thatimprove the lower bounds on minimum distance are presented. Furthermore, with new 2-generatorQC codes and Construction X, 2 new improved binary linear codes are obtained. With the standardconstruction techniques, another 16 new binary linear codes that improve the lower bound on theminimum distance have also been obtained.

A general algebraic method for decoding all types of binary cyclic codes is presented. It is shown that such a method can correct t=[(d-1)/2] errors, where d is the true minimum distance of the given cyclic code. The key idea behind this decoding technique is a systematic application of the algorithmic procedures of Grobner bases to obtain the error-locator polynomial L(z). The discussion begins from a set of syndrome polynomials F and the ideal T(F) generated by F. It is proved here that the process of transforming F to the normalized reduced Grobner basis of I(F) with respect to the purely lexicographical ordering automatically converges to L(z). Furthermore, it is shown that L(z) can be derived from any normalized Grobner basis of I(F) with respect to any admissible total ordering. To illustrate this new approach, the procedures for decoding certain BCH codes and quadratic residue codes are demonstrated. >

An elliptic semiplane is a λ-fold of a symmetric 2-(v,k,λ) design, where parallelism is transitive. We prove existence and uniqueness of a 3-fold cover of a 2-(15,7,3) design, and give several constructions. Then we prove that the automorphism group is 3.Alt(7). The corresponding bipartite graph is a minimal graph with valency 7 and girth 6, which has automorphism group 3.Sym(7). A polynomial with real coefficients is called formally positive if all of the coefficients are positive. We conjecture that the determinant of a matrix appearing in the proof of the van der Waerden conjecture due to Egorychev is formally positive in all cases, and we prove a restricted version of this conjecture. This is closely related to a problem concerning a certain generalization of Latin rectangles. Let ω be a primitive nth root of unity over GF(2), and let m_i(x) be the minimal polynomial of ω^i. The code of length n = 2^r-1 generated by m_1(x)m_t(x) is denoted C^t_r. We give a recursive formula for the number of codewords of weight 4 in C^11_r for each r.

Binary cyclic codes achieve good error correction performance and allow the implementation of very simpleencoder and decoder circuits. Among them, BCH codesrepresent a very important class of t-error correcting codes, with known structural properties and error correction capability. Decoding of binary cyclic codes is often accomplished through hard-decision decoders, although it is recognized that softdecision decoding algorithms can produce significant coding gain with respect to hard-decision techniques. Several approaches have been proposed to implement iterative soft-decision decoding of binary cyclic codes. We study the technique based on “extended parity-check matrices”, and show that such method is not suitable for high rates or long codes. We propose a new approach, based on “reduced parity-check matrices” and “spread parity-check matrices”, that can achieve better correction performance in many practical cases, without increasing the complexity.