Infinite families of $t$-designs from a type of five-weight codes

TL;DR

This paper constructs infinite families of 2- and 3-designs from binary linear codes with five weights, expanding the methods for deriving combinatorial designs from coding theory.

Contribution

It introduces a novel approach to generate infinite families of t-designs from a specific class of five-weight binary linear codes.

Findings

Number of designs is exponential in odd m.

Block sizes vary widely.

Constructs both 2- and 3-designs from codes.

Abstract

It has been known for a long time that $t$-designs can be employed to construct both linear and nonlinear codes and that the codewords of a fixed weight in a code may hold a $t$-design. While a lot of progress in the direction of constructing codes from $t$-designs has been made, only a small amount of work on the construction of $t$-designs from codes has been done. The objective of this paper is to construct infinite families of $2$-designs and $3$-designs from a type of binary linear codes with five-weights. The total number of $2$-designs and $3$-designs obtained in this paper are exponential in any odd $m$ and the block size of the designs varies in a huge range.