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

Abstract

Symbol-pair code is a new coding framework which is proposed to correct errors in the symbolpair read channel. In particular, maximum distance separable (MDS) symbol-pair codes are a kind of symbol-pair codes with the best possible error-correction capability. Employing cyclic and constacyclic codes, we construct three new classes of MDS symbol-pair codes with minimum pairdistance five or six. Moreover, we find a necessary and sufficient condition which ensures a class of cyclic codes to be MDS symbol-pair codes. This condition is related to certain property of a special kind of linear fractional transformations. A detailed analysis on these linear fractional transformations leads to an algorithm, which produces many MDS symbol-pair codes with minimum pair-distance seven.

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].…”

“…10] for the case of simple-root constacyclic codes. Since then symbol-pair distances over constacyclic codes are an interesting topic to study and under scrutiny in series of papers like [6], [10], [12]- [14], [19]. Constacyclic codes play a significant role in coding theory because of their rich algebraic structure and practical implementations.…”

b-Symbol Distance of Constacylic Codes of Length ps Over Fp m + uFp m

“…10] for simple-root constacyclic codes. Many researchers scrutinized symbol-pair distances over constacyclic codes since then in [11], [14], [16]- [18], [21] over many years.…”

Maximum distance separable codes for b-symbol read channels

“…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.…”

“…(6) there exists an MDS (n, 2b) q b-symbol code for q being a prime power, b ≥ 5, n ≥ 2b and b|n; (7) there exists an MDS ( q b+1 −1 q−1 , 2b + 1) q b-symbol code for q being prime power and any b ≥ 4. The family (4) indicates that a linear MDS b-symbol codes over F q with b = 3, d 3 = 6 or b = 4, d 4 = 8 exists for any length n, n ≥ 2b.…”