On self-dual MDS codes

Abstract

We consider the problem for which lengths a self-dual MDS code over F <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</inf> exists.We show that for q = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> , there are self-dual MDS codes for all even lengths up to 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> . Furthermore, self-dual MDS codes of length q + 1 over F <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</inf> exist for all odd prime powers q. Additionally, we present some new self-dual MDS codes.