On the Classification of MDS Codes

Abstract

A q-ary code of length n, size M , and minimum distance d is called an (n, M, d)q code. An (n, q k , n − k + 1)q code is called a maximum distance separable (MDS) code. In this work, some MDS codes over small alphabets are classified. It is shown that every7} is equivalent to a linear code with the same parameters. This implies that the (6, 5 4 , 3)5 code and the (n, 7 n−2 , 3)7 MDS codes for n ∈ {6, 7, 8} are unique. The classification of one-errorcorrecting 8-ary MDS codes is also finished; there are 14, 8, 4, and 4 equivalence classes of (n, 8 n−2 , 3)8 codes for n = 6, 7, 8, 9, respectively. One of the equivalence classes of perfect (9,8 7 , 3)8 codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction.

New MDS Self-Dual Codes From Generalized Reed—Solomon Codes

“…First of all, MDS codes achieve optimal parameters that allow correction of maximal number of errors for a given code rate. Study of various properties of MDS codes, such as classification [15], [20] of MDS codes, non-Reed-Solomon MDS codes [21], balanced MDS codes [6], lowest density MDS codes [4], [17] and existence of MDS codes [7], has been the center of the area. In addition, MDS codes are closely connected to combinatorial design and finite geometry [18,Chapters 11 and 14].…”

Further results on the classification of MDS codes

“…The nonexistence of pairs of mutually orthogonal Latin squares of order 6 implies the nonexistence of nontrivial 6-ary MDS codes with d ≥ 3. For q = 5, 7 all MDS codes with d ≥ 3, except the (4, 2) 7 codes, are equivalent to linear codes [10]. For q = 5, they are unique, which follows from the uniqueness in terms of the notion of equivalence of linear codes [6].…”

“…For q = 5, they are unique, which follows from the uniqueness in terms of the notion of equivalence of linear codes [6]. For q = 7, 8, the MDS codes with d = 3 were classified in [10,11]. For q = 7, some classification results exist for codes with d > 3 in terms of different notions of equivalence for linear codes; see for example [9].…”

On Robust Colorings of Hamming-Distance Graphs

“…The existence of (unrestricted) MDS codes is a longstanding open problem for most parameters. We refer to the literature on details about the conjecture; see for instance [10] by Kokkala et al and the references therein. We summarize the relation between existence of MDS codes and the chromatic number as follows.…”