BCH Codes Of Length Research Articles

The Dual Codes of Several Classes of 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.

New Quantum BCH Codes of Length n = r(q2 − 1)

New Quantum BCH Codes of Length n = r(q2 − 1)

Designed distances and parameters of new LCD BCH codes over finite fields

Let $\mathbb {F}_{q}$ be the finite field of q elements and n = qm − 1 with m a positive integer. In this paper we construct a class of BCH and LCD BCH codes of length n over $\mathbb {F}_{q}$ and investigate their dimensions and designed distance. Our results show that the designed distances of BCH and LCD BCH codes in this paper are larger than those in [11, Theorems 7, 10, 18, and 22]. It is viewed as a generalized result of [11].

Two families of BCH codes and new quantum codes

In this paper, two families of non-narrow-sense (NNS) BCH codes of lengths $$n=\frac{q^{2m}-1}{q^2-1}$$ and $$n=\frac{q^{2m}-1}{q+1}$$ ( $$m\ge 3)$$ over the finite field $$\mathbf {F}_{q^2}$$ are studied. The maximum designed distances $$\delta ^\mathrm{new}_\mathrm{max}$$ of these dual-containing BCH codes are determined by a careful analysis of properties of the cyclotomic cosets. NNS BCH codes which achieve these maximum designed distances are presented, and a sequence of nested NNS BCH codes that contain these BCH codes with maximum designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these NNS BCH codes. The new quantum codes presented here include many classes of good quantum codes, which have parameters better than those constructed from narrow-sense BCH codes, negacyclic and constacyclic BCH codes in the literature.

A New Powerful Scheme Based on Self Invertible Stabilizer Multiplier Permutation to Find the Minimum Distance for large BCH Codes

In this paper, we present the powerful scheme ZSISMP (Zimmermann Self Invertible Stabilizer Multiplier Permutation) to attack the hardness of the minimum distance search problem of BCH codes. This scheme consists in evaluating the minimum distance of the reduced dimension sub code fixed by a Self Invertible Stabilizer Multiplier Permutation by Zimmermann algorithm. The proposed scheme ZSISMP is validated on all BCH codes of known minimum distance. A comparison with several known powerful techniques proves its efficiency in giving more accurate results in short time. The use of this efficient local search had yield to determine the error correcting capability of many BCH codes of length 1023 and 4095.

Improved Constructions for Nonbinary Quantum BCH Codes

In this work, we present two improved constructions for nonbinary quantum BCH codes of lengths \(n=\frac {q^{4}-1}{2}\) and \(n=\frac {q^{4}-1}{q-1}\), where q is an odd prime power. Moreover, the constructed quantum BCH codes have parameters better than those obtained from other known constructions.

Some Hermitian Dual Containing BCH Codes and New Quantum Codes

Let q= 3l+ 2 be a prime power. Maximal designed distances of imprimitive Hermitian dual containing q 2 -ary narrow-sense (NS) BCH codes of length n = (q 6−1) 3 and n = 3(q 2 − 1)(q 2 + q+ 1) are determined. For each given n, non-narrow-sense (NNS) BCH codes which achieve such maximal designed distances are presented, and a series of NS and NNS BCH codes are constructed and their parameters are computed. Consequently, many families of q-ary quantum BCH codes are derived from these BCH codes. Some of these quantum BCH codes constructed from NNS BCH codes have better parameters than those quantum BCH codes available in the literature, and some others are new ones.

NEW QUANTUM CODES CONSTRUCTED FROM A CLASS OF IMPRIMITIVE BCH CODES

By a careful analysis on cyclotomic cosets, the maximal designed distance δnew of narrow-sense imprimitive Euclidean dual containing q-ary BCH code of length [Formula: see text] is determined, where q is a prime power and l is odd. Our maximal designed distance δnew of dual containing narrow-sense BCH codes of length n improves upon the lower bound δmax for maximal designed distances of dual containing narrow-sense BCH codes given by Aly et al. [IEEE Trans. Inf. Theory53 (2007) 1183]. A series of non-narrow-sense dual containing BCH codes of length n, including the ones whose designed distances can achieve or exceed δnew, are given, and their dimensions are computed. Then new quantum BCH codes are constructed from these non-narrow-sense imprimitive BCH codes via Steane construction, and these new quantum codes are better than previous results in the literature.

Coding Schemes for Noiseless and Noisy Asynchronous CDMA Systems

Novel coding schemes for noiseless and noisy asynchronous code division multiple access (A-CDMA) systems are presented in this paper. The schemes use Wu-Chang spreading code, block interleaver, and synchronizable channel codes to support A-CDMA communications with random delays. For the noiseless case, each active user generates one of M messages, M = ⌊(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T-D</sup> -1)/(D + 1)⌋ , and a (T-D)-stage MLSR encoder encodes the message into a codeword of length T. Given a maximum random delay of D chips, the receiver can decode messages of all active users without ambiguity. For the noisy case, extended Bose-Caldwell cyclic codes are used to encode messages. The extended cyclic codes provide error correction and delay recovery effectively. By using a BCH code of length <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> as the cyclic code, the extended code can correct up to τ ≤ ⌊n/4⌋ errors and recover a maximum random delay of n -1 chips.

The divisibility modulo 24 of Kloosterman sums on [formula omitted], m odd

In a previous paper, we studied the cosets of weight 4 of binary extended 3-error-correcting BCH codes of length 2 m (where m is odd). We expressed the number of codewords of weight 4 in such cosets in terms of exponential sums of three types, including the Kloosterman sums K ( a ) , a ∈ F ∗ . In this paper, we derive some congruences which link Kloosterman sums and cubic sums. This allows us to study the divisibility of Kloosterman sums modulo 24. More precisely, if we know the traces of a and of a 1 / 3 , we are able to evaluate K ( a ) modulo 24 and to compute the number of those a giving the same value of K ( a ) modulo 24.

On Cosets of Weight 4 of Binary BCH Codes with Minimum Distance 8 and Exponential Sums

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.

On minimum Lee weights of Hensel lifts of some binary BCH codes

Motivated by the paper of Calderbank, McGuire, Kumar, and Helleseth (see ibid., vol.42, no.1, p.217-26, Jan. 1996) we prove the following result: for any given positive integer l/spl ges/3, the minimum Lee weights of Hensel lifts (to Z/sub 4/) of binary primitive BCH codes of length 2/sup m/-1 and designed distance 2/sup l/-1 is just 2/sup l/-1 when (a) m can be divided by l or (b) m is sufficiently large. For Hensel lifts of binary primitive BCH codes of arbitrary designed distance /spl delta//spl ges/4, we also prove that their minimum Lee weight d/sub L//spl les/2([log/sub 2//spl delta/]+1)-1 when m is sufficiently large. Moreover, a result about minimum Lee weights of certain Z/sub 4/ codes defined by Galois rings, which is similar to the result in Calderbank et al., is proved.

Finding short and light codewords in BCH codes

A new algorithm is presented for finding codewords that are of low weight ("light") and low degree ("short") in narrow-sense binary BCH codes of length n=2/sup m/-1 and minimum weight d=2/sup t/-1, for use in serial implementations. This improves on some earlier results by Roth and Scroussi (see ibid., vol.34, p.593-6, May 1998), and extends the range of parameters to include cases of interest for long CRC (cyclic redundancy codes) error-detection schemes.

The Length of Primitive BCH Codes with Minimal Covering Radius

It is proved that the covering radius of a primitive binary BCH code of length q-1 and designed distance 2t+1, where \[ q=2^m>[(2t-3)(2t-1)!]^2, \] is exactly 2t-1 (the minimum value possible). The bound for q is significantly lower than the one obtained by O. Moreno and C. J. Moreno [9].

Long binary narrow-sense BCH codes are normal

Let C be the binary narrow-sense BCH code of length n=(2/sup m/-1)/N and designed distance 2t+1, where m is the order of 2 module n. Using characters of finite fields and a theorem of Weil, and results of Vladut-Skorobogatov (1989) and Lang-Weil (1954) we prove that the code C is normal in the non-primitive case N>1 if 2/sup m//spl ges/4(2tN)/sup 4t+2/, and in the primitive case N=1 if m/spl ges/m/sub 0/, where the constant m/sub 0/ depends only on t. >

Cyclic codes over Z/sub 4/, locator polynomials, and Newton's identities

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock (1972) and Preparata (1968) codes that can be very simply constructed as binary images, under the Gray map, of linear codes over Z/sub 4/ that are defined by means of parity checks involving Galois rings. This paper describes how Fourier transforms on Galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes. These methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of Z/sub 4/. Linear codes with length 2/sup m/ (m, odd) and size 2(2/sup m+1/-5m-2). The Gray image of the code of length 32 is the best (64, 2/sup 37/) code that is presently known. This paper also determines the exact minimum Lee distance of the linear codes over Z/sub 4/ that are obtained from the extended binary two- and three-error-correcting BCH codes by Hensel lifting. The Gray image of the Hensel lift of the three-error-correcting BCH code of length 32 is the best (64, 2/sup 32/) code that is presently known. This code also determines an extremal 32-dimensional even unimodular lattice.

A new procedure for decoding cyclic and BCH codes up to actual minimum distance

The paper presents a new procedure for decoding cyclic and BCH codes up to their actual minimum distance. It generalizes the Peterson decoding procedure and the procedure of Feng and Tzeng (1991) using nonrecurrent syndrome dependence relations. For a code with actual minimum distance d to correct up to t=[(d-1)/2] errors, the procedure requires a (2t+1)/spl times/(2t+1) syndrome matrix with known syndromes above the minor diagonal and unknown syndromes and their conjugates on the minor diagonal. In contrast to previous procedures, this procedure is primarily aimed at solving for the unknown syndromes instead of determining an error-locator polynomial. Decoding is then accomplished by determining the error vector as the inverse Fourier transform of the syndrome vector (S/sub 0/, S/sub 1/, S/sub n-1/). The authors show that with this procedure, all binary cyclic and BCH codes of length >

Weight distributions of cosets of two-error-correcting binary BCH codes, extended or not

Let B be the binary two-error-correcting BCH code of length 2/sup m/-1 and let B/spl circ/ be the extended code of B. We give formal expressions of weight distributions of the cosets of the codes B/spl circ/ only depending on m. We can then deduce the weight distributions of the cosets of B. When m is odd, it is well known that there are four distinct weight distributions for the cosets of B. So our main result is about the even case. Camion, Courteau, and Montpetit (see ibid., vol.38, no.7, p.1353, 1992) observe that for the lengths 15, 63, and 255 there are eight distinct weight distributions. We prove that this property holds for the codes B/spl circ/ and B for all even m.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On computing the weight spectrum of cyclic codes

Two deterministic algorithms of computing the weight spectra of binary cyclic codes are presented. These algorithms have the lowest known complexity for cyclic codes. For BCH codes of lengths 63 and 127, several first coefficients of the weight spectrum in number sufficient to evaluate the bounded distance decoding error probability are computed. >

A fast algorithm to determine the burst-correcting limit of cyclic or shortened cyclic codes

A novel fast algorithm is developed for computing the burst-correcting limit of a cyclic or shortened cyclic code from the parity-check polynomial of the cyclic code. The algorithm is similar to the algorithm of H.J. Matt and J.L. Massey (1980) which, up to now, has been the most efficient method for determining the burst-correcting limit of a cyclic code, but is based on apolarity of binary forms instead of linear complexity. The running times of implementations in C of both algorithms on an IBM RISC System/6000 are compared for several binary cyclic codes of practical interest. A table of the burst-correcting limit of primitive binary BCH codes of length up to 1023 is included. >