New constructions of MDS symbol-pair codes

Abstract

Motivated by the application of high-density data storage technologies, symbol-pair codes are proposed to protect against pair-errors in symbol-pair channels, whose outputs are overlapping pairs of symbols. The research of symbol-pair codes with the largest minimum pair-distance is interesting since such codes have the best possible error-correcting capability. A symbol-pair code attaining the maximal minimum pair-distance is called a maximum distance separable (MDS) symbol-pair code. In this paper, we focus on constructing linear MDS symbol-pair codes over the finite field F q . We show that a linear MDS symbol-pair code over F q with pair-distance 5 exists if and only if the length n ranges from 5 to q 2 + q + 1. As for codes with pair-distance 6, length ranging from 6 to q 2 +1, we construct linear MDS symbol-pair codes by using a configuration called ovoid in projective geometry. With the help of elliptic curves, we present a construction of linear MDS symbol-pair codes for any pair-distance d + 2 with length n satisfying 7 ≤ d + 2 ≤ n ≤ q + ⌊2 √ q⌋ + δ(q) − 3, where δ(q) = 0 or 1.

Maximum distance separable codes for b-symbol read channels

“…Actually, we can obtain MDS (n, 5) q 2-symbol codes for q being a prime power with length n ranging from 5 to q 2 + q + 1 according to Lemma 3.2. We do not present this as a theorem since the result has been covered by our former paper [5]. Besides providing a new proof of the result, the discussion in Lemma 3.4 will also help in the proof of Lemma 3.9.…”

“…Cassuto and Blaum [1] first proposed a new coding framework for symbol-pair read channels. For a complete comprehension of the fruitful results on this topic, please refer to [1,2,3,4,5,6,7,9] and the references therein. Recently, Yaakobi et al [10] generalized the coding framework for symbol-pair read channels to that for b-symbol read channels, where the read operation is performed as a consecutive sequence of b > 2 symbols.…”

MDS Symbol-Pair Repeated-Root Constacylic Codes of Prime Power Lengths Over $\mathbb F_{p^m}+ u \mathbb F_{p^m}$

“…Maximum distance separable (MDS) codes are optimal in the sense that they have the highest possible error-detecting and error-correcting capability for given code length and code size. This encourages numerous investigations regarding construction of MDS codes with respect to symbol-pair metric like [7], [8], [15], [18], [19].…”

A new class of MDS symbol-pair codes

“…The one is based on linear codes with certain properties, especially, MDS codes [3,4] or constacyclic (cyclic) codes [5, 10, 15-17, 20, 21]. The other is based on interleaving techniques [3,4], Eulerian graphs [3,4], projective geometry [6] and algebraic geometry codes over elliptic curves [6]. On the other hand, it is also a hot topic to calculate the weight distribution for a linear code.…”