Recursive construction of a class of cyclic difference families
By a \(\widetilde T(v,k,k - 1)\)-difference family in the group \(Z_v\) of residue classes modulo \(v\) is meant a collection of cyclic ordered \(k\)- subsets (blocks) of \(Z_v\) giving each nonzero difference of \(Z_v\) at any fixed distance \(\rho = 1,2, \dots, k - 1\) (cyclically) exactly once. We make use of the notion \(\widetilde T(v,k,k - 1)\)-difference families in \(Z_v\) if the blocks of the family define a partition of the nonzero residues of \(Z_v\). We are interested in \(\widetilde T(v,k,k - 1)\)-difference family in \(Z_v\) which are closed with respect to multiplication by any nonzero residue of \(Z_v\). For such difference families we prove a recursive theorem of product type when \(k |(p - 1)\) for each prime divisor \(p\) of the number \(vw\) and \(k \geq 3\). As a consequence of the theorem we prove the effectiveness of search in the resulting \(\widetilde T(vw,k,k - 1)\)-difference family in \(Z_{vw}\) for a cyclic group of order \(k\) in \(Z_{vw}\) and a generator of it. Closely connected with the concept of \(\widetilde T(v,k,k - 1)\)- difference family is Mendelsohn's concept of perfect design \((v,k, 1)\). Applying the group of automorphisms \(Z_v\) to the blocks of a \(\widetilde T(v,k,k - 1)\)-difference family we get Mendelsohn's perfect scheme \((v,k,1)\). By a \((v,d)\)-difference matrix in \(Z_v\) is meant a \(d \times v\) rectangular matrix with elements from \(Z_v\) such that among the corresponding differences of elements of any two rows each element of \(Z_v\) occurs exactly once. By a BIB-design with parameters \((v,k, \lambda)\) is meant a collection of \(k\)-subsets of a \(v\)-set containing each 2-subset exactly \(\lambda\) times. From the existence of a \(\widetilde T(v,k,k - 1)\)-difference family in \(Z_v\) it follows that there exists a BIB-design with parameters \((v,k,k - 1)\) having \(Z_v\) as a group of automorphisms.
scientific article; zbMATH DE number 846480