A class of almost MDS codes

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Abstract

MDS codes and almost MDS (AMDS) codes are special classes of linear codes, and have important applications in communications, data storage, combinatorial theory, and secrete sharing. The objective of this paper is to present a class of AMDS codes from some BCH codes and determine their parameters. It turns out the proposed AMDS codes are distance-optimal and dimension-optimal locally repairable codes. The parameters of the duals of this class of AMDS codes are also discussed.

Introduction

In coding theory, the Singleton bound is an upper bound on the size of an arbitrary linear code with length n, size M, and minimum distance d. For an [n,k,d] code, the Singleton bound says that dnk+1. If this equality holds, i.e., d=nk+1, then the code is called an maximum distance separable (MDS) code. If d=nk, then the code is called an almost MDS (AMDS for short) code [10]. A code is said to be a near MDS (NMDS for short) code if the code and its dual code both are AMDS. MDS codes achieve optimal parameters that allow correction of maximal number of errors for a given code rate [6]. MDS and AMDS codes have important applications in communications, data storage, combinatorial theory, and secret sharing [12].

In many cases, BCH codes are the best linear codes. In the past ten years, a lot of progress on the study of BCH codes has been made (see, for example, [7][8][9][11][16]). Recently, Ding and Tang studied a class of BCH codes C(q,n,δ,h) with q=3m, length n=q+1, design distance δ=3 and h=1, and proved this class of BCH codes are NMDS codes [2]. They also obtained infinite families of 3-designs from the proposed NMDS codes. Inspired by the work of Ding and Tang, the objective of this paper is to present a class of AMDS codes from the BCH codes C(q,q+1,3,4) and determine their parameters. It turns out the proposed AMDS codes are distance-optimal and dimension-optimal locally repairable codes, which have important applications in distributed storage systems and receive a lot of attention in recent years [14][15][5][1][13]. The parameters of the duals of this class of AMDS codes are also discussed.

The rest of this paper is organized as follows. In Section 2, we present some preliminaries on cyclic codes, BCH codes, AMDS codes, NMDS codes and locally repairable codes. In Section 3, we propose a new class of AMDS codes from BCH codes over GF(3m) with m being odd. Section 4 is devoted to some concluding remarks.

Section snippets

Cyclic codes, BCH codes, almost MDS codes and near MDS codes

An [n,k,d] code C over GF(q) is said to be cyclic if (c0,c1,,cn1)C implies (cn1,c0,c1,,cn2)C. By identifying any vector (c0,c1,,cn1)GF(q)n with c0+c1x+c2x2++cn1xn1GF(q)[x]/(xn1), any code C of length n over GF(q) corresponds to a subset of the quotient ring GF(q)[x]/(xn1). A linear code C is cyclic if and only if the corresponding subset in GF(q)[x]/(xn1) is an ideal of the ring GF(q)[x]/(xn1).

Note that every ideal of GF(q)[x]/(xn1) is principal. Let C=g(x) be a cyclic

A class of AMDS codes

Throughout this section, let q=3m, where m is odd. In this section, we consider a new class of BCH code C(q,q+1,3,4) over GF(3m) and its dual, and prove that C(q,q+1,3,4) is an AMDS code.

The following lemma will be useful in the sequel whose proof is straightforward.

Lemma 3.1

Let x,y,zGF(q2). Then|x4y4z4x5y5z5x5y5z5|=x5y5z5[(y9z9)(z10x10)(x9z9)(z10y10)].

We will also need the following lemma.

Lemma 3.2

Let q=3m with m being odd. Let Uq+1 denote the set of all (q+1)-th roots of unity in GF(q2). Suppose

Summary and concluding remark

The contribution of this paper was the study of the BCH code C(q,q+1,3,4) in Section 3, where the parameters of the code were determined, and a bound on the minimum distance of the dual of the code C(q,q+1,3,4) was given. Furthermore, the code C(q,q+1,3,4) over GF(q) is a distance-optimal and dimension-optimal LRC. It would be possible to obtain more AMDS codes from some other BCH codes. Finally, we invite the reader to prove or disprove Conjecture 3.6.

Acknowledgements

The authors would like to thank the Editor and the anonymous Reviewers for giving us invaluable comments and suggestions that greatly improved the quality of this paper. This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 62071397 and 11971395, and also supported in part by projects of central government to guide local scientific and technological development under Grant No. 2021ZYD0001.

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