Linear Codes from Simplicial Complexes over $\mathbb{F}_{2^n}$

TL;DR

This paper constructs new linear codes over finite fields using simplicial complexes, analyzes their parameters, and identifies infinite families of optimal and minimal codes, advancing coding theory with geometric combinatorics.

Contribution

It introduces a novel method of constructing linear codes from simplicial complexes over ^n and determines their parameters, including conditions for minimality.

Findings

Constructed five infinite families of distance optimal codes.

Established relations between codeword weights and LFSR sequences.

Provided conditions under which these codes are minimal.

Abstract

In this article we mainly study linear codes over $\mathbb{F}_{2^n}$ and their binary subfield codes. We construct linear codes over $\mathbb{F}_{2^n}$ whose defining sets are the certain subsets of $\mathbb{F}_{2^n}^m$ obtained from mathematical objects called simplicial complexes. We use a result in LFSR sequences to illustrate the relation of the weights of codewords in two special codes obtained from simplical complexes and then determin the parameters of these codes. We construct fiveinfinite families of distance optimal codes and give sufficient conditions for these codes to be minimal.