Designs, permutations, and transitive groups

TL;DR

This paper investigates the structure of $t$-designs in the symmetric group, deriving bounds on their size for $t=1,2$ and showing the non-existence of tight 2-designs through combinatorial and linear programming methods.

Contribution

It introduces new lower bounds for $t$-designs in the symmetric group and proves the non-existence of tight 2-designs using power moment and linear programming techniques.

Findings

Lower bounds for $t=1,2$ designs established.

Linear programming bounds for general $n,t$ derived.

Tight 2-designs do not exist for the parameters considered.

Abstract

A notion of $t$-designs in the symmetric group on $n$ letters was introduced by Godsil in 1988. In particular $t$-transitive sets of permutations form a $t$-design. We derive special lower bounds for $t=1$ and $t=2$ by a power moment method. For general $n,t$ we give a %linear programming lower bound . For $n\ge 4$ and $t=2,$ this bound is strong enough to show a lower bound on the size of such $t$-designs of $n(n-1)\dots (n-t+1),$ which is best possible when sharply $t$-transitive sets of permutations exist. This shows, in particular, that tight $2$-designs do not exist.