New MDS Self-Dual Codes from Generalized Reed-Solomon Codes

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). Furthermore, we prove that there exists a q-ary MDS self-dual code of length n if q = r 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. Index Terms Self-dual codes, MDS codes, Generalized Reed-Solomon codes.