Linear codes using simplicial complexes

TL;DR

This paper constructs and analyzes linear codes over finite fields using simplicial complexes, establishing relations between codes and their subfield codes, and identifies infinite families of distance optimal codes.

Contribution

It introduces a novel construction of linear codes from simplicial complexes and derives conditions for optimality and minimality, including five infinite families of distance optimal codes.

Findings

Established a relation between $C_{D}$ and $C_{D}^{(2)}$ using a generator matrix.

Obtained five infinite families of distance optimal codes.

Provided sufficient conditions for codes to be minimal.

Abstract

Certain simplicial complexes are used to construct a subset $D$ of $\mathbb{F}_{2^n}^m$ and $D$, in turn, defines the linear code $C_{D}$ over $\mathbb{F}_{2^n}$ that consists of $(v\cdot d)_{d\in D}$ for $v\in \mathbb{F}_{2^n}^m$. Here we deal with the case $n=3$, that is, when $C_{D}$ is an octanary code. We establish a relation between $C_{D}$ and its binary subfield code $C_{D}^{(2)}$ with the help of a generator matrix. For a given length and dimension, a code is called distance optimal if it has the highest possible distance. With respect to the Griesmer bound, five infinite families of distance optimal codes are obtained, and sufficient conditions for certain linear codes to be minimal are established.