The linear codes of t-designs held in the Reed-Muller and Simplex codes

TL;DR

This paper investigates the structure of linear codes derived from t-designs embedded in Reed-Muller and Simplex codes, providing new theoretical insights and discussing open problems in the field.

Contribution

It introduces general theory for linear codes of t-designs within Reed-Muller and Simplex codes, advancing understanding of their properties and relationships.

Findings

Established connections between t-designs and linear codes in Reed-Muller and Simplex codes.

Presented general theoretical framework for analyzing these linear codes.

Outlined open problems for future research in the area.

Abstract

A fascinating topic of combinatorics is $t$-designs, which have a very long history. The incidence matrix of a $t$-design generates a linear code over GF$(q)$ for any prime power $q$, which is called the linear code of the $t$-design over GF$(q)$. On the other hand, some linear codes hold $t$-designs for some $t \geq 1$. The purpose of this paper is to study the linear codes of some $t$-designs held in the Reed-Muller and Simplex codes. Some general theory for the linear codes of $t$-designs held in linear codes is presented. Open problems are also presented.