Orderly generation of half regular symmetric designs via Rahilly families of pre-difference sets (Q1976883)

Rahilly families of pre-difference sets were introduced and their properties were investigated by \textit{A. J. Rahilly}, \textit{C. E. Praeger}, \textit{A. P. Street} and \textit{D. E. Bryant} [Australas. J. Comb. 8, 1-26 (1993; Zbl 0795.05021)] in order to construct symmetric 2-designs with \(v=2w\) even and having nice automorphism groups. A Rahilly family of pre-difference sets is defined as follows: Let \(v\), \(k\) and \(\lambda\) be positive integers which are assumed as parameters of a nontrivial symmetric \(2\)-design. Let \(G\) be a group of order \(w\) and let \(\Delta_{ij}\) for \(\{i,j\}= \{1,2\}\) be a subset of \(G\) of size \(k_{ij}\) such that \(k_{1j}+ k_{2j}= k\). Then \(\Delta= \{\Delta_{ij}: i,j\in\{1,2\}\}\) is called a Rahilly family of pre-difference sets for \(G\) with parameters \(v, k, \lambda\) if: (a) for each \(g\neq 1\in G\) and \(i\in\{1,2\}\) there is an integer \(\lambda_i(g)\) such that \(0\leq\lambda_i(g)\leq\lambda\) and \(g\) can be written exactly \(\lambda_i(g)\) times as \(cd^{-1}\) with \(c,d\in \Delta_{ii}\) and exactly \(\lambda-\lambda_i(g)\) times as \(ef^{-1}\) with \(e, f\in\Delta_{ij}\), where \(\{i,j\}= \{1,2\}\); (b) for each \(g\in G\) and \(\{i,j\}= \{1,2\}\) there is an integer \(\lambda_{ij}(g)\) such that \(0\leq \lambda_{ij}(g)\leq\lambda\) and \(g\) can be written exactly \(\lambda_{ij}(g)\) times as \(cd^{-1}\) with \(c\in\Delta_{ii}\) and \(d\in\Delta_{ji}\) and exactly \(\lambda- \lambda_{ij}(g)\) times as \(ef^{-1}\), with \(e\in\Delta_{ij}\) and \(f\in \Delta_{jj}\). Now the authors invent two backtracking algorithms called canonical transversal and all Rahilly families, which, using results of the paper mentioned above, enable them to determine all putative parameter sets for Rahilly families of groups of orders up to 110. Furthermore, for \(G= Z_2\times Z_3\times Z_3\) they classify all Rahilly families, and eleven half-regular symmetric \((36, 15, 6)\) designs arise. In particular, they show that there are half-regular symmetric designs whose full automorphism groups do not contain regular subgroups.