Using rational idempotents to show Turyn’s bound is sharp

Abstract

Davis, Dillon, and Jedwab all showed the existence of difference sets in groups $${C_{2^{r+2}}\times C_{2^{r}}}$$ . Turyn's bound had previously shown that abelian 2-groups with higher exponents could not admit difference sets. We give a new construction technique that utilizes character values, rational idempotents, and tiling structures to produce Hadamard difference sets in the group $${C_{2^{r+2}}\times C_{2^{r}}}$$ to replicate the result.