Construction of codes over a commutative non-unital ring from simplicial complexes and their applications

2026-06-15

AI Summary

This work investigates the construction of linear codes with desirable properties over finite non-chain rings by leveraging simplicial complexes with multiple maximal elements and the defining-set approach. It introduces, for the first time, such multi-maximal-element simplicial complexes into code constructions over these rings and systematically establishes sufficient conditions for obtaining minimal codes, optimal codes, self-orthogonal codes, and locally recoverable codes. Through Gray maps and subfield-subcode analysis, the study yields several families of divisible codes, projective few-weight codes, and locally recoverable codes with small locality. Furthermore, it characterizes the minimal access structures of secret sharing schemes based on dual minimal codes and explicitly constructs several strongly regular graphs from two-weight codes, thereby extending the applications of coding theory in cryptography and combinatorial design.

Abstract

In this article, we investigate the construction of linear codes over a finite ring $\mathcal{S}$, where $\mathcal{S}$ is taken to be an extension of a commutative non-unital ring $I$ of order $p^2$. Our approach is based on the defining set method. The defining sets considered in this work are derived from general simplicial complexes that may contain multiple maximal elements. We determine the parameters of these codes over $\mathcal{S}$ and study their Gray images. We also study the corresponding subfield-like codes. We show that these Gray image codes and subfield-like codes produce several families of divisible codes. Furthermore, we establish sufficient conditions under which these codes are minimal, optimal, and self-orthogonal. As applications of our results, we obtain several families of projective few-weight codes, and locally recoverable codes with small locality. We also study the minimal access structures of secret-sharing schemes associated with the duals of these minimal codes. Moreover, we construct several families of strongly regular graphs from projective two-weight codes and determine their parameters explicitly.

Problem

• linear codes
• commutative non-unital ring
• simplicial complexes
• Gray image
• few-weight codes

Innovation

• simplicial complexes
• commutative non-unital ring
• linear codes
• Gray image
• minimal codes