On a relation between a cyclic relative difference set associated with the quadratic extensions of a finite field and the Szekeres difference sets

Abstract

Letq≡ 3 (mod 4) be a prime power and put$n = \frac{{q - 1}}{2}$. We consider a cyclic relative difference set with parametersq ²−1,q, 1,q−1 associated with the quadratic extension GF(q²)/GF((q). The even part and the odd part of the cyclic relative difference set taken modulon are$2 - \left\{ {n;\frac{{n + 1}}{2};\frac{{n + 1}}{2}} \right\}$ supplementary difference sets. Moreover it turns out that their complementary subsets are identical with the Szekeres difference sets. This result clarifies the true nature of the Szekeres difference sets. We prove these results by using the theory of the relative Gauss sums.