Block-transitive t-(k^2,k,\lambda) designs associated to two dimensional projective special linear groups

TL;DR

This paper classifies certain highly symmetric combinatorial designs with parameters related to projective special linear groups, identifying specific conditions under which these designs exist.

Contribution

It establishes the existence and structure of t-(k^2,k, extlambda) designs with block-transitive automorphism groups related to PSL(2,q), specifically for q=8.

Findings

q=8 is the only case where such designs exist

The design is a 2-(36,6, extlambda) with specific lambda values

Automorphism group is linked to PSL(2,8)

Abstract

This paper investigates block-transitive automorphism groups of t-(k^2,k,\lambda) designs. Let D be a non-trivial t-(k^2,k,\lambda) design, G \leq \Aut(D) be block-transitive with X\unlhd G\leq \Aut(X), where X = PSL(2,q)(q\geq4). Then q = 8 and D is a 2-(36,6,\lambda) design with \lambda \in \{2,6,9,12,18,36\}.