Quaternary codes with new parameters from two-generator simplicial complexes

TL;DR

This paper constructs new infinite families of quaternary codes using simplicial complexes, determines their weight distributions, and identifies codes with optimal or improved parameters, including projective and minimal binary codes.

Contribution

It introduces a novel method for constructing quaternary codes from two-generator simplicial complexes and characterizes when their Gray images are linear, leading to new optimal codes.

Findings

At least 32 new or improved quaternary codes identified.

Six projective quaternary codes with best-known parameters reported.

Infinite families of Griesmer and minimal binary codes established.

Abstract

In this article, we construct infinite families of quaternary (that is, over the ring $\mathbb{Z}_4$) $\mathcal{C}_{D}$-codes, where the defining set $D$ is derived utilizing a two-generator simplicial complex, and determine their Lee weight distributions. As a result, we find at least 32 new or improved quaternary linear codes as per the database \cite{aydin2022updated} of best-known quaternary codes, including codes from a Plotkin-optimal family. We also report 6 projective quaternary linear codes with best-known parameters that might outperform the currently reported best-known codes due to their projectivity. Further, we establish necessary and sufficient conditions for their Gray image to be linear, which in turn gives an infinite family of Griesmer codes and several infinite families of minimal binary linear codes.