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.

In this paper, we present three new classes of q-ary quantum MDS codes utilizing generalized Reed-Solomon codes satisfying Hermitian self-orthogonal property. Among our constructions, the minimum distance of some q-ary quantum MDS codes can be bigger than q 2 + 1. Comparing to previous known constructions, the lengths of codes in our constructions are more flexible.

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