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

TL;DR

This paper introduces new classes of near MDS self-dual and isodual codes over finite fields, expanding the known constructions of MDS and near MDS self-dual codes using extended negacyclic duadic codes.

Contribution

It generalizes MDS self-dual codes to near MDS variants and provides new constructions of these codes over finite fields.

Findings

Constructed many new near MDS isodual codes from extended negacyclic duadic codes.

Derived numerous MDS Euclidean self-dual codes from existing MDS Euclidean self-dual codes.

Extended the theory to near MDS Hermitian self-dual codes with new constructions.

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.