# Some new designs with prescribed automorphism groups

## TL;DR

This paper proves the existence of several previously unknown simple combinatorial designs with specific parameters, expanding the catalog of known designs and analyzing the conditions for their existence.

## Contribution

It establishes the existence of new simple designs with various parameters and investigates the set of lambda values for which such designs exist.

## Key findings

- Existence of simple 2-(55,10,4) design
- Existence of simple 3-(20,5,4) and 3-(21,7,30) designs
- Existence of simple 4-(15,5,2), 4-(16,8,45) designs, and 5-(16,7,10), 5-(17,8,40) designs

## Abstract

We establish the existence of simple designs with parameters $2$-$(55,10,4)$, $3$-$(20,5,4)$, $3$-$(21,7,30)$, $4$-$(15,5,2)$, $4$-$(16,8,45)$, $5$-$(16,7,10)$, and $5$-$(17,8,40)$, which have previously been unknown. For the corresponding $t$, $v$, and $k$, we study the set of all $\lambda$ for which simple $t$-$(v,k,\lambda)$ designs exist.