Quaternary linear codes and related binary subfield codes

TL;DR

This paper explores the relationship between quaternary linear codes and their binary subfield codes, constructing new codes, analyzing their weight distributions, and identifying optimal and near-optimal codes.

Contribution

It provides explicit relationships between quaternary and binary codes, constructs new codes via simplicial complexes, and discovers at least nine new quaternary linear codes.

Findings

Derived explicit relationships between quaternary and binary codes.

Constructed new codes with known weight distributions.

Identified infinite families of optimal and near-optimal codes.

Abstract

In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and defining sets. Second, we construct quaternary linear codes via simplicial complexes and determine the weight distributions of these codes. Third, the weight distributions of the binary subfield codes of these quaternary codes are also computed by employing the general characterization. Furthermore, we present two infinite families of optimal linear codes with respect to the Griesmer Bound, and a class of binary almost optimal codes with respect to the Sphere Packing Bound. We also need to emphasize that we obtain at least 9 new quaternary linear codes.