Improved Lower Bounds on the Minimum Distances of the Dual Codes of Primitive Narrow-Sense BCH Codes

As a special subclass of cyclic codes, BCH codes have wide applications in communication and storage systems. A BCH code of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> is always relative to an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-th primitive root of unity <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> in an extension field of <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>, and is called a dually-BCH code if its dual is also a BCH code relative to the same <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula>. The question as to whether a BCH code is a dually-BCH code is in general very hard to answer. In this paper, an answer to this question for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes is given. Sufficient and necessary conditions in terms of the designed distances <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> will be presented to ensure that these BCH codes are dually-BCH codes. In addition, the parameters of the primitive narrow-sense BCH codes and their dual codes are investigated. Some lower bounds on minimum distances of the dual codes of primitive and projective narrow-sense BCH codes are developed. Especially for binary primitive narrow-sense BCH codes, the new bounds on the minimum distances of the dual codes improve the classical Sidel&#x2019;nikov bound, and are also better than the Carlitz and Uchiyama bound for large designed distances <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>. The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent. As a byproduct, the parameters of some subclasses of cyclic codes are also investigated.

In this work, we introduce a method by which it is established that; how a sequence of non-primitive BCH codes can be obtained by a given primitive BCH code. For this, we rush to the out of routine assembling technique of BCH codes and use the structure of monoid rings instead of polynomial rings. Accordingly, it is gotten that there is a sequence $\{C_{b^{j}n}\}_{1\leq j\leq m}$, where $b^{j}n$ is the length of $C_{b^{j}n}$, of non-primitive binary BCH codes against a given binary BCH code $C_{n}$ of length $n$. Matlab based simulated algorithms for encoding and decoding for these type of codes are introduced. Matlab provides built in routines for construction of a primitive BCH code, but impose several constraints, like degree $s$ of primitive irreducible polynomial should be less than $16$. This work focuses on non-primitive irreducible polynomials having degree $bs$, which go far more than 16.

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. >

In this work, we introduce a method by which it is established that; how a sequence of non-primitive BCH codes can be obtained by a given primitive BCH code. For this, we rush to the out of routine assembling technique of BCH codes and use the structure of monoid rings instead of polynomial rings. Accordingly, it is gotten that there is a sequence $\{C_{b^{j}n}\}_{1\leq j\leq m}$, where $b^{j}n$ is the length of $C_{b^{j}n}$, of non-primitive binary BCH codes against a given binary BCH code $C_{n}$ of length $n$. Matlab based simulated algorithms for encoding and decoding for these type of codes are introduced. Matlab provides built in routines for construction of a primitive BCH code, but impose several constraints, like degree $s$ of primitive irreducible polynomial should be less than $16$. This work focuses on non-primitive irreducible polynomials having degree $bs$, which go far more than 16.

In this paper, we present constructions of primitive and non-primitive BCH codes using monoid rings over the local ring [Formula: see text], with [Formula: see text]. We show that there exist two sequences [Formula: see text] and [Formula: see text] of non-primitive BCH codes (over [Formula: see text] and [Formula: see text], respectively) against primitive BCH codes [Formula: see text] of length [Formula: see text] and [Formula: see text] (over [Formula: see text] and [Formula: see text]), respectively. A technique is developed in an innovative way that enables the data path to shift instantaneously during transmission via the coding schemes of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. The selection of the schemes is subject to the choice of better code rate or better error-correction capability of the code. Finally, we present a decoding procedure for BCH codes over Galois rings, which is also used for the decoding of BCH codes over Galois fields, based on the modified Berlekamp–Massey algorithm.

Primitive BCH codes with symbols from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(q)</tex> and designed distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> have parameter values \begin{align} \text{block length} &= n = q^m - 1 \\ \text{check symbols/block} &= r \leq m(d - 1) \end{align} where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> is any positive integer. For many nonbinary BCH codes (called maximally redundant codes), the maximum number of check symbols per block is required, i.e. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r = m(d - 1)</tex> . Conditions whereby a primitive nonbinary BCH code is maximally redundant are discussed. It is shown that a class of codes exists, with symbols from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(q)</tex> , based upon doubly lengthened Reed-Solomon codes over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(q^m)</tex> , having parameter values \begin{align} \text{block length} &= n = m(q^m + 1) \\ \text{check symbols/block} &= r = m(d - 1) \\ \text{designed distance} &= d \end{align} where again <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> is any positive integer. Thus this class of codes extends the block length of maximally redundant codes by a multiplicative factor exceeding <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> , while retaining the same designed distance and same number of check symbols.

We study coset weight distributions of binary primitive (narrow-sense) BCH codes of length n = 2 m (m odd) with minimum distance 8. In the previous paper [1], we described coset weight distributions of such codes for cosets of weight j = 1, 2, 3, 5, 6. Here we obtain exact expressions for the number of codewords of weight 4 in terms of exponential sums of three types, in particular, cubic sums and Kloosterman sums. This allows us to find the coset distribution of binary primitive (narrow-sense) BCH codes with minimum distance 8 and also to obtain some new results on Kloosterman sums over finite fields of characteristic 2.

Dimensions of three types of BCH codes over [formula omitted

In this paper, we discuss the problem on the construction of asymmetric quantum codes [[n, k, dz/dx]]4 from primitive narrow sense BCH codes. Take advantage of dual containing BCH codes of length n = 4m -- 1 where m = 2l and m = 2l + 1, we construct two families of new asymmetric quantum codes with dz &#x2265; &#x03B4;max + 1, where &#x03B4;max is the maximal designed distance of dual containing narrow sense BCH code. Our asymmetric quantum codes are new ones, and are not included in the known literature.

Usually spectra (weight distributions) of primitive binary BCH codes are supposed to approximate binomial weight distributions well for a wide range of code rates and code lengths. It is shown that for any fixed code rate R<1 spectra of long (N/spl rarr//spl infin/) primitive binary BCH codes cannot approximate the binomial distribution at all.

A new step-by-step decoder for double-error-correcting primitive binary BCH codes in normal basis is presented in this paper. This decoder uses a new technique that can determine whether the checked bit is in error or not. This technique also can directly decode any bit in a received vector without knowing the number of errors and also without temporarily changing any received bit. We also transformed the syndrome values S 1 and S 3 from conventional basis to normal basis, since in normal basis, computing a cube in GF(2 m ) is faster and simpler than in conventional basis. Moreover, owing to the simple and regular decoding procedure, the new algorithm is suitable for VLSI implementation. In this paper, we use the new method to implement the (15,7) double-error-correcting primitive binary BCH codes in normal basis. The decoding speed (or data rate) of the new decoder is 2776kbits−1 for a 5.55MHz clock rate and m = 7, which is faster than a conventional decoder.

The dimension and minimum distance of two classes of primitive BCH codes

A lower bound on the number of uncorrectable errors of weight half the minimum distance is derived for binary linear codes satisfying some condition. The condition is satisfied by some primitive BCH codes, extended primitive BCH codes, Reed-Muller codes, and random linear codes. The bound asymptotically coincides with the corresponding upper bound for Reed-Muller codes and random linear codes. By generalizing the idea of the lower bound, a lower bound on the number of uncorrectable errors for weights larger than half the minimum distance is also obtained, but the generalized lower bound is weak for large weights. The monotone error structure and its related notion larger half and trial set, which are introduced by Helleseth, Kloslashve, and Levenshtein, are mainly used to derive the bounds.

Due to wide applications of BCH codes, the determination of their minimum distance is of great interest. However, this is a very challenging problem for which few theoretical results have been reported in the last four decades. Even for the narrow-sense primitive BCH codes, which form the most well studied subclass of BCH codes, there are very few theoretical results on the minimum distance. In this paper, we present new results on the minimum distance of narrow-sense primitive BCH codes with special Bose distance. We prove that for a prime power $q$, the $q$-ary narrow-sense primitive BCH code with length $q^m-1$ and Bose distance $q^m-q^{m-1}-q^i-1$, where $\frac{m-2}{2} \le i \le m-\lfloor \frac{m}{3} \rfloor-1$, has minimum distance $q^m-q^{m-1}-q^i-1$. This is achieved by employing the beautiful theory of sets of quadratic forms, symmetric bilinear forms, and alternating bilinear forms over finite fields, which can be best described using the framework of association schemes.

It was shown earlier that for a punctured Reed-Muller (RM) code or a primitive BCH code, which contains a punctured RM code of the same minimum distance as a large subcode, the state complexity of the minimal trellis diagrams is much greater than that for an equivalent code obtained by a proper permutation of the bit positions. The problem of finding a permutation of the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is related to the generalized Hamming weight hierarchy of a code, and it is shown that, for RM codes, the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to V.K. Wei (1991). The state complexity of the trellis diagram for the extended and permuted (64, 24) BCH code is discussed. >