Symmetric $(2^k-1,2^{k-1},2^{k-2})$-designs which are $(2^{k-1}-1)$-pyramidal over abelian groups

TL;DR

This paper classifies symmetric combinatorial designs with specific parameters that exhibit a high degree of symmetry, focusing on those that are pyramidal over abelian groups, expanding understanding of their structure.

Contribution

The paper fully determines all symmetric $(2^k-1,2^{k-1},2^{k-2})$-designs that are $(2^{k-1}-1)$-pyramidal over abelian groups, providing a complete classification.

Findings

All such designs are explicitly characterized.

The designs are shown to have a specific automorphism group structure.

The classification applies for all integer values of k.

Abstract

A design is called $t$-pyramidal when it has an automorphism group which fixes $t$ points and acts sharply transitively on the remaining points. We determine all symmetric $(2^k-1,2^{k-1},2^{k-2})$-designs which are $(2^{k-1}-1)$-pyramidal over abelian groups.