Hypergraph encoding set systems and their linear representations

TL;DR

This paper explores the encoding of set systems and t-designs using hypergraphs, linking their structure to linear group actions and algebraic geometry codes over finite fields.

Contribution

It introduces a novel hypergraph encoding framework for t-designs and connects this to the representation theory of linear groups and algebraic geometry code constructions.

Findings

Hypergraph encoding captures the structure of t-designs over finite fields.

The approach relates design theory to linear group representations.

Connections to algebraic geometry codes are established.

Abstract

We study $t$-designs of parameters $(n,k,\lambda)$ over finite fields as group divisible designs and set systems admitting a transitive action of a linear group encoded in an hypergraph $G$ whose vertex set of size $n$ is partitioned into sets of size $k$ in such a way that every $t$-subset is contained in at least $\lambda$ subsets of $G$. We relate the problem to the representation theory of the general linear group $\GL(n,\mathbb{F}_{q})$ and the constructions of AG codes over finite fields.