Infinite families of optimal and minimal codes over rings using simplicial complexes

TL;DR

This paper constructs infinite families of optimal and minimal codes over rings using simplicial complexes, characterizes their parameters, and derives numerous linear codes over finite fields with desirable properties.

Contribution

It introduces new infinite families of codes over rings via simplicial complexes and analyzes their parameters, weight distributions, and optimality, including self-orthogonality conditions.

Findings

Complete Lee weight distributions for certain codes.

Many codes satisfy Ashikhmin-Barg's minimality condition.

Two families of distance-optimal codes over finite fields.

Abstract

In this paper, several infinite families of codes over the extension of non-unital non-commutative rings are constructed utilizing general simplicial complexes. Thanks to the special structure of the defining sets, the principal parameters of these codes are characterized. Specially, when the employed simplicial complexes are generated by a single maximal element, we determine their Lee weight distributions completely. Furthermore, by considering the Gray image codes and the corresponding subfield-like codes, numerous of linear codes over $\mathbb{F}_q$ are also obtained, where $q$ is a prime power. Certain conditions are given to ensure the above linear codes are (Hermitian) self-orthogonal in the case of $q=2,3,4$. It is noteworthy that most of the derived codes over $\mathbb{F}_q$ satisfy the Ashikhmin-Barg's condition for minimality. Besides, we obtain two infinite families of distance-optimal codes over $\mathbb{F}_q$ with respect to the Griesmer bound.