Unequal-field generalization of the construction

Adapt the paper’s cyclotomic construction to construct partial difference sets in the additive group (F_{q_1} × F_{q_2}, +) with q_1 and q_2 distinct prime powers.

Background

The paper’s main construction uses products of finite fields whose orders are powers of a common base q, specifically groups of the form (F_{q^{m\ell}} × F_{q^{m(\ell+1)}}, +). The authors point to Ott’s broader framework for products of finite fields and explicitly propose adapting their approach to the case of two finite fields with unequal underlying parameters q_1 ≠ q_2.