Maximum Distance Separable Codes for Symbol-Pair Read Channels

Abstract

We study (symbol-pair) codes for symbol-pair read channels introduced recently by Cassuto and Blaum (2010). A Singleton-type bound on symbol-pair codes is established and infinite families of optimal symbol-pair codes are constructed. These codes are maximum distance separable (MDS) in the sense that they meet the Singleton-type bound. In contrast to classical codes, where all known -ary MDS codes have length , we show that -ary MDS symbol-pair codes can have length . In addition, we completely determine the existence of MDS symbol-pair codes for certain parameters.Index Terms-Codes for magnetic storage, maximal distance separable, Singleton-type bound, symbol-pair read channels.

New constructions of MDS symbol-pair codes

“…The construction of MDS symbolpair codes is interesting since the codes have the best pair-error correcting capability for fixed length and dimension. The authors in [4] made use of interleaving and graph theoretic concepts as well as combinatorial configurations to construct MDS symbol-pair codes. Kai et al [8] constructed MDS symbol-pair codes from cyclic and constacyclic codes.…”

“…Classical MDS codes are MDS symbol-pair codes [4] and other known families of MDS (n, d) q symbol-pair codes are shown in Table 1. 4 q ≥ 2 n ≥ 2 [4] even prime power n ≤ q + 2 [4] odd prime 5 ≤ n ≤ 2q + 3 [4] 5 prime power n|q 2 − 1, n > q + 1 [8] prime power n = q 2 + q + 1 [8] prime power, q ≡ 1 (mod 3) n = q 2 +q+1 3 [8] 6 prime power n = q 2 + 1 [8] odd prime power n = q 2 +1 2 [8] 7 odd prime n = 8 [4] In this paper, we present new constructions of linear MDS symbol-pair codes over the finite field F q and obtain the following three new families:…”

Constructions of maximum distance separable symbol-pair codes using cyclic and constacyclic codes

“…In general, there are two ways to construct MDS symbol-pair codes. The first one is direct construction using linear codes with appropriate properties, such as MDS codes [4], as well as cyclic and constacyclic codes [7]. The second way is recursive construction employing the interleaving technique [4,5], the Eulerian graph [4,5,7] and other combinatorial configurations [4,5].…”

“…In particular, we focus on the construction of (n, d p ) q MDS symbol-pair code whose minimum pair-distance d p is small. The known parameters of (n, d p ) q MDS symbol-pair codes with small d p are the following ones: a) q ≥ 2, n ≥ 2, d p ∈ {2, 3} [4], b) q ≥ 2, n ≥ 4, d p = 4 [4], c1) q is an even prime power, n ≤ q + 2, d p = 5 [4], c2) q is an odd prime, 5 ≤ n ≤ 2q + 3, d p = 5 [4], c3) q is a prime power, n | q 2 − 1, n > q + 1, d p = 5 [7], c4) q is a prime power, n = q 2 + q + 1, d p = 5 [7], c5) q ≡ 1 (mod 3) is a prime power, n = q 2 +q+1 3 , d p = 5 [7], d1) q is a prime power, n = q 2 + 1, d p = 6 [7], d2) q is an odd prime power, n = q 2 +1 2 , d p = 6 [7], e) q is an odd prime, n = 8, d p = 7 [4].…”

List Decodability of Symbol-Pair Codes

“…A code C with minimum pair distance d P can uniquely correct t pair errors if and only if d P ≥ 2t + 1 see [2]. Hence, it is desirable to keep minimum pair distance d P as large as possible for a symbol-pair code with fixed n. It has been shown [4] that an (n, M, d P ) q -symbol-pair code C must obey the following version of the Singleton bound.…”