On the existence of block-transitive combinatorial designs

TL;DR

This paper proves that non-trivial block-transitive Steiner t-designs do not exist for t=7 or larger, using classification of finite 3-homogeneous groups and simple groups.

Contribution

It establishes the non-existence of certain highly symmetric combinatorial designs for t≥7, resolving an open problem in the field.

Findings

No non-trivial block-transitive Steiner 7-designs exist.

The proof relies on classification of finite 3-homogeneous groups.

Connects group theory with combinatorial design theory.

Abstract

Block-transitive Steiner $t$-designs form a central part of the study of highly symmetric combinatorial configurations at the interface of several disciplines, including group theory, geometry, combinatorics, coding and information theory, and cryptography. The main result of the paper settles an important open question: There exist no non-trivial examples with $t=7$ (or larger). The proof is based on the classification of the finite 3-homogeneous permutation groups, itself relying on the finite simple group classification.