New MDS self-dual codes over finite fields $\F_{r^2}$

TL;DR

This paper develops new theoretical methods to construct six classes of MDS self-dual codes over finite fields, significantly expanding the known code families and increasing the proportion of known codes.

Contribution

The paper introduces six new classes of MDS self-dual codes over finite fields, enhancing the existing theory and expanding the known code families.

Findings

Constructed six new classes of MDS self-dual codes.

Achieved the largest known ratio (over 57%) of known to possible MDS self-dual codes.

Also constructed new MDS self-orthogonal and almost self-dual codes.

Abstract

MDS self-dual codes have nice algebraic structures and are uniquely determined by lengths. Recently, the construction of MDS self-dual codes of new lengths has become an important and hot issue in coding theory. In this paper, we develop the existing theory and construct six new classes of MDS self-dual codes. Together with our constructions, the proportion of all known MDS self-dual codes relative to possible MDS self-dual codes generally exceed 57\%. As far as we know, this is the largest known ratio. Moreover, some new families of MDS self-orthogonal codes and MDS almost self-dual codes are also constructed.