Power difference set problem

Determine for which divisors \(\ell\) of \(q-1\) the multiplicative subgroup of \(\ell\)-th powers in a finite field \(F_q\) forms a difference set with respect to the additive group of \(F_q\).

Background

A power difference set is the multiplicative subgroup $K$ of $\ell$-th powers in $F_q^\times$ that is simultaneously a difference set in the additive group of $F_q$. The paper explains that this problem originated in work of Paley and Chowla and has been studied through cyclotomic numbers, explicit formulas, and cyclotomic matrices.

The general existence problem remains unresolved, particularly as the index $\ell$ grows. The paper develops matrix-theoretic necessary and sufficient conditions and spectral and determinant constraints, but does not settle the general classification.