A new family of relative difference sets in \(2\)-groups (Q1963148)

Let \(G\) be an abelian group of order \(2^{ab}\) and \(N\) a subgroup of \(G\) of order \(2^b\). A subset \(R\) of \(G\) is a semiregular relative difference set if \(|R|=2^a\) and every element \(g\in G\setminus N\) has \(2^{a-b}\) representations \(r-r'\), \(r, r'\in R\). Quite a few constructions are known if \(b\leq a/2\). In case \(b>a/2\) there were only two constructions known: a (trivial) one using projections of difference sets with \(a=b\), and a nontrivial one with \(a=2b-1\); see \textit{Y. Q. Chen, D. K. Ray-Chaudhuri} and \textit{Q. Xiang} [Constructions of partial difference sets and relative difference sets using Galois rings. II. J. Comb. Theory, Ser. A 76, No. 2, 179-196 (1996; Zbl 0859.05021)]. It is called the ``square root problem'' to find more series with \(b>a/2\). Using building sets [see \textit{J. A. Davis} and \textit{J. Jedwab}, A unifying construction for difference sets, J. Comb. Theory, Ser. A 80, No. 1, 80, 13-78 (1997; Zbl 0884.05019)] the authors are able to construct infinitely many new semiregular relative difference sets including new examples with \((a,b)=(4,3)\). This is the only case where the parameters of the new series satisfy \(b>a/2\).