A subfield-based construction of optimal linear codes over finite fields

TL;DR

This paper introduces four new families of linear codes over finite fields constructed from subfield complements, achieving optimality and specific weight distributions, with potential applications in coding theory.

Contribution

It presents novel constructions of optimal linear codes from subfield complements, providing explicit optimality criteria and characterizing their weight distributions.

Findings

Constructed four families of optimal linear codes.

Identified many distance-optimal codes with few weights.

Most codes are self-orthogonal or minimal.

Abstract

In this paper, we construct four families of linear codes over finite fields from the complements of either the union of subfields or the union of cosets of a subfield, which can produce infinite families of optimal linear codes, including infinite families of (near) Griesmer codes. We also characterize the optimality of these four families of linear codes with an explicit computable criterion using the Griesmer bound and obtain many distance-optimal linear codes. In addition, we obtain several classes of distance-optimal linear codes with few weights and completely determine their weight distributions. It is shown that most of our linear codes are self-orthogonal or minimal which are useful in applications.