Euclidean and Hermitian self-dual MDS codes over large finite fields

The first author constructed new extremal binary self-dual codes (IEEE Trans. Inform. Theory 47 (2001) 386-393) and new self-dual codes over GF(4) with the highest known minimum weights (IEEE Trans. Inform. Theory 47 (2001) 1575-1580). The method used was to build self-dual codes from a given self-dual code of a smaller length. In this paper, we develop a complete generalization of this method for the Euclidean and Hermitian self-dual codes over finite fields GF(q). Using this method we construct many Euclidean and Hermitian self-dual MDS (or near MDS) codes of length up to 12 over various finite fields GF(q), where q = 8, 9, 16, 25, 32, 41, 49, 53, 64, 81, and 128. Our results on the minimum weights of (near) MDS self-dual codes over large fields give a better bound than the Pless-Pierce bound obtained from a modified Gilbert-Varshamov bound.