  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 $p^2$.
