Abstract
Reversible difference sets have been studied extensively by many people. Dillon showed that reversible difference sets existed in groups \({(C_{2^{r}})^{2}}\) and C4. Davis and Polhill showed the existence of DRAD difference sets in the groups \({(C_{2^{r}})^{2}}\) for \({r\geq 2}\) and also for the group C4. This paper gives a construction technique utilizing character values, rational idempotents, and tiles to produce both reversible and DRAD Hadamard difference sets in the group \({C_{2^{r}} \times C_{2^{r}}}\) for \({r\geq 2}\) and in C4. We also show necessary conditions for both reversible and DRAD difference sets in abelian 2-groups.
Highlights
Let G be a group of order v written multiplicatively and let D be a k-subset of G
The support of a group ring element is the set of group elements with nonzero coefficients
We say that a group ring element X has been shifted by g if it is multiplied by g
Summary
Let G be a group of order v written multiplicatively and let D be a k-subset of G. If D is a (v, k, λ) difference set, the group ring element D = G − 2D is the Hadamard difference set transform of D. We work exclusively with the Hadamard difference set transforms of difference sets For this reason, in the remainder of the paper we use the terminology “difference set” for. The group ring element D is a (4m2, 2m2 − m, m2 − m) difference set if and only if D D(−1) = 4m2 and D has coefficients of ±1. DRAD difference sets can be thought of as a combination of a skew-symmetric element D − N added to a subgroup N of order 2m.
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