New MDS self-dual codes over finite fields of odd characteristic

Abstract

In this paper, we produce new classes of MDS self-dual codes via (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. Among our constructions, there are many MDS self-dual codes with new parameters which have never been reported. For odd prime power q with q square, the total number of lengths for MDS self-dual codes over F q presented in this paper is much more than those in all the previous results.

A Unified Approach to Construct MDS Self-Dual Codes via Reed-Solomon Codes

“…We present simple proofs with the approach illustrated in Section II. For more complete list of known MDS self-dual codes we refer to the table in [3] and [15].…”

“…In [3], {ξ i : 1 ≤ i ≤ t} is taken as a subset of θ r−1 for q = r 2 . Firstly we should determine that how many elements ξ 1 , · · · , ξ t in θ r−1 can be taken such that the cosets ξ i C (1 ≤ i ≤ t) are distinct.…”

“…With above preparation we introduce the following result which appear in [3] but with slight different statement. Proof.…”

New Construction of Maximum Distance Separable (MDS) Self-Dual Codes over Finite Fields

“…The problem is completely solved for the case where q is even [ 8 ]. Many MDS self-dual codes over finite fields of odd characteristics were constructed [ 9 , 10 , 11 , 12 , 13 , 14 ].…”

“…In [ 12 ], Ladad, Liu and Luo produced more classes of MDS self-dual codes based on [ 11 ] and [ 14 ]. In [ 9 ], based on the [ 11 , 12 , 14 ] more new parameter MDS self-dual codes were presented. Based on the method raised in [ 9 ], we present some classes of MDS self-dual codes.…”