On symbol-pair weight distribution of MDS codes and simplex codes over finite fields
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On symbol-pair weight distribution of MDS codes and simplex codes over finite fields Junru Ma1 · Jinquan Luo1 Received: 26 April 2020 / Accepted: 17 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Symbol-pair code is a new coding framework proposed to combat pair-errors in symbolpair read channels. Remarkably, a classical maximum distance separable (MDS) code is also an MDS symbol-pair code. In this paper, we investigate the symbol-pair weight distribution of MDS codes and the symbol b-weight distribution of simplex codes over finite fields respectively. Surprisingly, one can derive that all the nonzero codewords of simplex codes have the same symbol b-weight and rearrangement of coordinates in simplex codes may induce different symbol-pair weights. Moreover, the symbol b-weight distribution of variation simplex codes over certain finite fields is determined. Keywords MDS symbol-pair codes · Symbol-pair weight · Symbol b-weight · MDS codes · Simplex codes Mathematics Subject Classification (2010) 11T71
1 Introduction Symbol-pair codes were first proposed in [1, 2] by Cassuto and Blaum to protect against pair-errors in symbol-pair read channels. It was shown in [3] that there exist symbol-pair codes with rates that are strictly higher than the best known codes in the Hamming metric. Shortly afterwards, Chee et al. [4, 5] established a Singleton-type bound on symbol-pair codes and constructed several classes of MDS symbol-pair codes using the known MDS codes with interleaving techniques. Later, Ding et al. [8] extended the Singleton-type bound to the b-symbol case and constructed MDS b-symbol codes based on projective geometry. Moreover, several works were contributed to the constructions of MDS symbol-pair codes in [6, 7, 9, 12, 14, 17–19]. Meanwhile, some decoding algorithms of symbol-pair codes Jinquan Luo
[email protected] Junru Ma [email protected] 1
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China
Cryptography and Communications
were proposed by various researchers in [15, 20–24] and the references therein. Recently, the symbol-pair distances of a class of repeated-root cyclic codes over Fpm were determined in [26]. Later, Dinh et al. [11, 13] successfully derived the symbol-pair distances for all constacyclic codes of length p s and p 2s over Fpm . More recently, the Hamming and symbolpair distances of all the repeated-root constacyclic codes of length ps over R = Fpm + uFpm were presented in [10]. However, there are few works on determining the symbol-pair distances of MDS symbol-pair codes. MDS symbol-pair codes are optimal in the sense that such codes attain maximal minimum symbol-pair distance and the classical MDS codes are MDS symbol-pair codes [4]. In this paper, we will investigate the symbol-pair distances of MDS codes. It should be noted that the symbol-pair weight distribution of MDS codes is determined. The contributions of this paper are
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