New constructions of NMDS self-dual codes

TL;DR

This paper introduces new methods for constructing NMDS self-dual codes over finite fields, significantly expanding the known parameters and providing effective approaches for their construction, especially over square fields.

Contribution

It presents a novel class of $q$-ary linear codes that are either MDS or NMDS, with a new approach to construct NMDS self-dual codes extending existing parameter ranges.

Findings

Constructed a large class of NMDS self-dual codes for square $q$.

Provided an effective construction method for NMDS self-dual codes.

Almost $q/8$ NMDS self-dual codes can be constructed for square $q$.

Abstract

Near maximum distance separable (NMDS) codes are important in finite geometry and coding theory. Self-dual codes are closely related to combinatorics, lattice theory, and have important application in cryptography. In this paper, we construct a class of $q$-ary linear codes and prove that they are either MDS or NMDS which depends on certain zero-sum condition. In the NMDS case, we provide an effective approach to construct NMDS self-dual codes which largely extend known parameters of such codes. In particular, we proved that for square $q$, almost $q/8$ NMDS self-dual $q$-ary codes can be constructed.