3-Designs from PSL(2,q) with cyclic starter blocks

TL;DR

This paper investigates the conditions under which the projective special linear group over a finite field can define specific 3-designs with cyclic starter blocks, establishing equivalences for certain parameter sets based on prime power congruences.

Contribution

It establishes new equivalences for the existence of specific 3-designs derived from PSL(2,q) with cyclic starters, linking design parameters to prime power congruences.

Findings

Existence of 3-(q+1,5,3) and 3-(q+1,10,18) designs for q ≡ 1 mod 20

Existence of 3-(q+1,13,33) and 3-(q+1,26,150) designs for q ≡ 1 mod 52

Conditions based on prime power congruences for design existence

Abstract

We consider when the projective special linear group over a finite field defines a $3$-design with a cyclic starter block. We will show that the equivalences of the existence of such $3$-$(q+1,5,3)$ and $3$-$(q+1,10,18)$ designs for a prime power $q\equiv 1\pmod{20}$, and $3$-$(q+1,13,33)$ and $3$-$(q+1,26,150)$ designs for a prime power $q\equiv 1\pmod{52}$, respectively.