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

Abstract

Both MDS and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, determining existence of q-ary MDS self-dual codes for various lengths has been investigated extensively. The problem is completely solved for the case where q is even. The current paper focuses on the case where q is odd. We construct a few classes of new MDS self-dual codes through generalized Reed-Solomon codes. More precisely, we show that for any given even length n we have a q-ary MDS code as long as q ≡ 1 mod 4 and q is sufficiently large (say q ≥ 4 n × n 2 ). Furthermore, we prove that there exists a q-ary MDS self-dual code of length n if q = r 2 and n satisfies one of the three conditions: (i) n ≤ r and n is even; (ii) q is odd and n − 1 is an odd divisor of q − 1; (iii) r ≡ 3 mod 4 and n = 2tr for any t ≤ (r − 1)/2.

Constructions of optimal LCD codes over large finite fields

“…. , C 4 ]A with parameters [16,4,12] Since there is no MDS linear code with parameters [16,4,13] over F 11 , this almost MDS code is optimal and has new parameters ever.…”

“…From the existence of MDS self-dual codes for example in [8,13] as well as from MDS self-orthgonal codes, we construct MDS LCD codes with certain lengths. We also generalize the constructions [20] of orthogonal matrices from prime fields to arbitrary finite fields and afterwards we give explicit constructions of LCD and MPLCD codes.…”

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

“…As a direct consequence of Theorem 2.2, we have Corollary 2.4. ( [9]) Suppose that r = p m and q = r 2 , where p is an odd prime number and m ≥ 1. Then for any even number n, n ∈ Σ(g, q) if 2 ≤ n ≤ r − 1 and n ∈ Σ(eg, q) if 4 ≤ n ≤ r + 1.…”

Explicit Constructions of MDS Self-Dual Codes

“…Grassl et al [10] constructed MDS codes of all lengths over F 2 m and of all highest possible length over finite fields of odd characteristics. Jin et al [13] proved the existence of MDS self-dual codes over F q in odd characteristic for q ≡ 1 (mod 4) and for q a square of a prime for some restricted lengths. Using the same technique developed in [13], more families of MDS self-dual codes have been constructed in [24,7].…”