The existence of pyramidal Steiner triple systems over abelian groups

TL;DR

This paper characterizes the existence of f-pyramidal Steiner triple systems over abelian groups for all f>3, completing previous results for smaller f values.

Contribution

It determines the spectrum of (f,v) pairs for which f-pyramidal STS(v) over abelian groups exist, using difference family constructions.

Findings

Complete characterization of f-pyramidal STS(v) over abelian groups for all f>3.

Constructed difference families relative to partial spreads.

Resolved existence questions for previously unknown parameter ranges.

Abstract

A Steiner triple system STS$(v)$ is called $f$-pyramidal if it has an automorphism group fixing $f$ points and acting sharply transitively on the remaining $v-f$ points. In this paper, we focus on the STSs that are $f$-pyramidal over some abelian group. Their existence has been settled only for the smallest admissible values of $f$, that is, $f=0,1,3$. In this paper, we complete this result and determine, for every $f>3$, the spectrum of values $(f,v)$ for which there is an $f$-pyramidal STS$(v)$ over an abelian group. This result is obtained by constructing difference families relative to a suitable partial spread.