Geometric Constructions of Set Systems and Their Hypergraph Representations

Abstract

We study t-designs of parameters (n, k, λ) over finite fields as group divisible designs and set systems admitting a transitive action of a linear group represented by an hypergraph G whose vertex set of size n is partitioned into sets of size k in such a way that every t sub set is contained in at least λ subsets of G. We relate the problem to the representation theory of the general linear group GL(n, Fq) and the constructions of AG codes over finite fields. As a byproduct we construct a RS code based encryption scheme