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

Both Maximum Distance Separable 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 the 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. This 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: 1) n ≤ r and n is even; 2) q is odd and n - 1 is an odd divisor of q - 1; and 3) r ≡ 3 mod 4 and n=2tr for any t ≤ (r - 1)/2.