# On solving isomorphism problems about 2-designs using block intersection numbers

## TL;DR

This paper addresses an isomorphism problem for 2-designs using block intersection numbers, providing partial solutions and insights into the limitations of the technique, with additional results on cyclotomic numbers and Galois rings.

## Contribution

It introduces a novel approach to isomorphism problems in 2-designs via block intersection analysis, extending understanding of Galois rings and cyclotomic numbers.

## Key findings

- Partial solutions to the isomorphism problem for 2-designs.
- Bounding techniques for block intersection numbers.
- Structural insights into Galois rings of characteristic p^2.

## Abstract

In this paper, we give a partial solution to a new isomorphism problem about $2$-$(v,k,k-1)$ designs from disjoint difference families in finite fields and Galois rings. Our results are obtained by carefully calculating and bounding some block intersection numbers, and we give insight on the limitations of this technique. Moreover, we present results on cyclotomic numbers and on the structure of Galois rings of characteristic $p^2$.