There exist Steiner systems $S(2,7,505)$, $S(2,7,589)$, and $S(2,8,624)$

TL;DR

This paper presents new Steiner systems that resolve several previously undecided cases in combinatorial design theory, specifically for block lengths 7 and 8.

Contribution

It provides explicit constructions of Steiner systems S(2,7,505), S(2,7,589), and S(2,8,624), solving multiple open problems.

Findings

Two Steiner systems S(2,7,505) constructed

Two Steiner systems S(2,7,589) constructed

Ten Steiner systems S(2,8,624) constructed

Abstract

In this note two Steiner systems $S(2,7,505)$, two Steiner systems $S(2,7,589)$, and ten Steiner systems $S(2,8,624)$ are presented. This resolves two of $21$ undecided cases for block designs with block length $7$, and one of $37$ cases for block designs with block length $8$, mentioned in Handbook of Combinatorial Designs.