On non-symmetric relative difference sets (Q949094)

An \((m,u,k,\lambda)\)\textit{-relative difference set} in a group \(G\) of order \(mu\) relative to a subgroup \(U\) of order \(u\) is a \(k\)-subset \(D\) of \(G\) with the property that the multiset of non-identity products \(d_1d_2^{-1}\), \(d_1,d_2 \in D \), contains each element of \(G \setminus U \) \(\lambda\) times, and contains no elements of \(U\). The \((m,u,k,\lambda)\)-relative difference set \(D\) is called \textit{symmetric} if \(D^{-1} = \{ d^{-1} \; | \; d \in D \} \) is also an \((m,u,k,\lambda)\)-relative difference set for the same group/subgroup pair. If the index \(m\) of \(U\) in \(G\) is equal to the size \(k\) of \(D\) then \(D\) is called \textit{semiregular}. Given a relative difference set \(D\), the associated incidence structure \(dev(D)\) with the point set \(G\) and the block set \( \{ Dg \; | \; g \in G \} \) is known to be an \((m,u,k,\lambda)\)\textit{-divisible design}. In the main result of the paper, the author shows that a semiregular relative difference set \(D\) is symmetric if and only if the dual of \(dev(D)\) is also a divisible design. This is followed by a recursive construction of an infinite family of semiregular non-symmetric relative difference sets. The construction uses a modification of Davis' recursive construction [\textit{J. A. Davis}, ``A note on products of relative difference sets'', Des. Codes Cryptography 1, No.2, 117--119 (1991; Zbl 0754.05014)] started from a non-symmetric \((12,3,12,4)\)-relative difference set.