New MDS or near MDS self-dual codes over finite fields

Abstract

The study of MDS self-dual codes has attracted lots of attention in recent years. There are many papers on determining existence of $q-$ary MDS self-dual codes for various lengths. There are not existence of $q-$ary MDS self-dual codes of some lengths, even these lengths $< q$. We generalize MDS Euclidean self-dual codes to near MDS Euclidean self-dual codes and near MDS isodual codes. And we obtain many new near MDS isodual codes from extended negacyclic duadic codes and we obtain many new MDS Euclidean self-dual codes from MDS Euclidean self-dual codes. We generalize MDS Hermitian self-dual codes to near MDS Hermitian self-dual codes. We obtain near MDS Hermitian self-dual codes from extended negacyclic duadic codes and from MDS Hermitian self-dual codes.