Solving isomorphism problems about 2-designs from disjoint difference families

TL;DR

This paper demonstrates that certain 2-designs derived from difference families in Galois rings and finite fields are nonisomorphic by analyzing their block intersection numbers, addressing a complex isomorphism problem.

Contribution

The paper proves nonisomorphism of 2-designs from Galois rings and finite fields using block intersection number analysis, advancing understanding of their structural differences.

Findings

Designs from Galois rings and finite fields are nonisomorphic.

Block intersection numbers distinguish the two classes of designs.

Addresses an open problem on isomorphism of these combinatorial structures.

Abstract

Recently, two new constructions of $(v,k,k-1)$ disjoint difference families in Galois rings were presented by Davis, Huczynska, and Mullen and Momihara. Both were motivated by a well-known construction of difference families from cyclotomy in finite fields by Wilson. It is obvious that the difference families in the Galois ring and the difference families in the finite field are not equivalent. A related question which is in general harder to answer is whether the associated designs are isomorphic or not. In our case, this problem was raised by Davis, Huczynska and Mullen. In this paper we show that the $2$-$(v,k,k-1)$ designs arising from the difference families in Galois rings and those arising from the difference families in finite fields are nonisomorphic by comparing their block intersection numbers.