Quaternary codes with new parameters from two-generator simplicial complexes

2026-05-14

AI Summary

This work addresses the construction of optimal quaternary linear codes by introducing a novel approach based on two-generator simplicial complexes. Specifically, it defines a linear code $\mathcal{C}_D$ over the ring $\mathbb{Z}_4$ via a defining set $D$, and investigates its Lee weight distribution together with the linearity conditions of its Gray image. This method represents the first systematic application of two-generator simplicial complexes to construct quaternary codes, yielding 32 new or improved optimal quaternary codes—including an infinite family achieving the Plotkin bound—six projective quaternary codes with leading parameters, and infinite families of Griesmer codes as well as minimal binary linear codes. These results substantially advance the current state-of-the-art in optimal code constructions.

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.

Problem

• quaternary codes
• simplicial complexes
• Lee weight distribution
• projective codes
• Gray image

Innovation

• quaternary codes
• simplicial complexes
• Lee weight distribution
• Gray image
• minimal binary linear codes