Certain binary minimal codes constructed using simplicial complexes

TL;DR

This paper constructs and analyzes binary minimal codes derived from simplicial complexes over a specific ring, providing new classes of optimal, few-weight codes with minimal and self-orthogonal properties.

Contribution

It introduces a novel construction of binary codes from simplicial complexes over a non-chain ring, including conditions for minimality and self-orthogonality, and presents infinite families of optimal codes.

Findings

Most codes are few-weight codes.

Conditions for minimality and self-orthogonality are established.

An infinite family of optimal codes is constructed.

Abstract

In this manuscript, we work over the non-chain ring $\mathcal{R} = \mathbb{F}_2[u]/\langle u^3 - u\rangle $. Let $m\in \mathbb{N}$ and let $L, M, N \subseteq [m]:=\{1, 2, \dots, m\}$. For $X\subseteq [m]$, define $\Delta_X:=\{v \in \mathbb{F}_2^m : \textnormal{Supp}(v)\subseteq X\}$ and $D:= (1+u^2)D_1 + u^2D_2 + (u+u^2)D_3$, an ordered finite multiset consisting of elements from $\mathcal{R}^m$, where $D_1\in \{\Delta_L, \Delta_L^c\}, D_2\in \{\Delta_M, \Delta_M^c\}, D_3\in \{\Delta_N, \Delta_N^c\}$. The linear code $C_D$ over $\mathcal{R}$ defined by $\{\big(v\cdot d\big)_{d\in D} : v \in \mathcal{R}^m \}$ is studied for each $D$. Further, we also consider simplicial complexes with two maximal elements in the above work. We study their binary Gray images and the binary subfield-like codes corresponding to a certain $\mathbb{F}_{2}$-functional of $\mathcal{R}$. Sufficient conditions for these binary linear codes to be minimal and self-orthogonal are obtained in each case. Besides, we produce an infinite family of optimal codes with respect to the Griesmer bound. Most of the codes obtained in this manuscript are few-weight codes.