# An infinite family of Steiner systems $S(2, 4, 2^m)$ from cyclic codes

## TL;DR

This paper introduces a novel infinite family of Steiner systems $S(2, 4, 2^m)$ derived from cyclic codes, expanding the known constructions and providing new combinatorial designs for specific parameters.

## Contribution

It presents the first coding-theoretic construction of an infinite family of Steiner systems $S(2, 4, v)$ for all $m
ot
ot ext{congruent to } 0 ext{ mod } 4$ with $m
eq 2$, specifically for $m ext{ mod } 4 = 2$ and $m extgreater 6$.

## Key findings

- Constructed an infinite family of Steiner systems $S(2, 4, 2^m)$ for $m ext{ mod } 4 = 2$ and $m extgreater 6$.
- Generated many infinite families of 2-designs as a by-product.
- First coding-theoretic construction of such Steiner systems.

## Abstract

Steiner systems are a fascinating topic of combinatorics. The most studied Steiner systems are $S(2, 3, v)$ (Steiner triple systems), $S(3, 4, v)$ (Steiner quadruple systems), and $S(2, 4, v)$. There are a few infinite families of Steiner systems $S(2, 4, v)$ in the literature. The objective of this paper is to present an infinite family of Steiner systems $S(2, 4, 2^m)$ for all $m \equiv 2 \pmod{4} \geq 6$ from cyclic codes. This may be the first coding-theoretic construction of an infinite family of Steiner systems $S(2, 4, v)$. As a by-product, many infinite families of $2$-designs are also reported in this paper.