Abstract
A major conjecture on the existence of abelian skew Hadamard difference sets is: if an abelian group $G$ contains a skew Hadamard difference set, then $G$ must be elementary abelian. This conjecture remains open in general. In this paper, we give a recursive construction for skew Hadamard difference sets in abelian (not necessarily elementary abelian) groups. The new construction can be considered as a result on the aforementioned conjecture: if there exists a counterexample to the conjecture, then there exist infinitely many counterexamples to it.
Highlights
Let G be an additively written group, let G∗ = G \ {0G}, and let D be a subset of G
Two difference sets D1 and D2 in an abelian group G are said to be equivalent if there is a group automorphism σ of G and an element b ∈ G such that σ(D1) = D2 + b
Let D be a skew Hadamard difference set in G, and let D + x = {y + x : y ∈ D} for x ∈ G
Summary
We assume the existence of 25 skew Hadamard difference sets in abelian groups of order q to construct a skew Hadamard difference set in an abelian group of order q5. We assume that each Gi contains some (not necessarily distinct) skew Hadamard difference sets Hi. We construct a skew Hadamard difference set D in G = G0 × G1 × · · · × Gn−1 so that D is a union of direct products of either {0Gi}, Gi, Hi or −Hi for 0 i n − 1 as in Example 3. The main point of our recursive construction is that the assumed abelian groups containing skew Hadamard difference sets are not necessarily the electronic journal of combinatorics 27(3) (2020), #3.36 elementary abelian. We apply our construction for small q’s and discuss about the inequivalence of resulting skew Hadamard difference sets
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