A new structure for difference matrices over abelian pp-groups

A difference matrix over a group is a discrete structure that is intimately related to many other combinatorial designs, including mutually orthogonal Latin squares, orthogonal arrays, and transversal designs. Interest in constructing difference matrices over $2$-groups has been renewed by the recent discovery that these matrices can be used to construct large linking systems of difference sets, which in turn provide examples of systems of linked symmetric designs and association schemes. We survey the main constructive and nonexistence results for difference matrices, beginning with a classical construction based on the properties of a finite field. We then introduce the concept of a contracted difference matrix, which generates a much larger difference matrix. We show that several of the main constructive results for difference matrices over abelian $p$-groups can be substantially simplified and extended using contracted difference matrices. In particular, we obtain new linking systems of difference sets of size $7$ in infinite families of abelian $2$-groups, whereas previously the largest known size was $3$.Comment: 27 pages. Discussion of new reference [LT04

On difference matrices of coset type

AbstractA (u,k;λ)-difference matrix H over a group U is said to be of coset type with respect to one of its rows, say w, whose entries are not equal, if it has the property that rw is also a row of H for any row r of H. In this article we study the structural property of such matrices with u(<k) a prime and show that u|λ and, moreover, H contains u (u,k/u;λ/u)-difference submatrices and is equivalent to a special kind of extension using them. Conversely, we also show that any set of u (u,k′;λ′)-difference matrices over U yields a (u,uk′;uλ′)-difference matrix of coset type over U

Concerning difference matrices

Several new constructions for difference matrices are given. One class of constructions uses pairwise balanced designs to develop new difference matrices over the additive group of GF(q). A second class of constructions gives difference matrices over groups whose orders are not (necessarily) prime powers

Concerning Difference Matrices

Several new constructions for difference matrices are given. One class of constructions uses pairwise balanced designs to develop new difference matrices over the additive group of GF(q). A second class of constructions gives difference matrices over groups whose orders are not (necessarily) prime powers. 1 Introduction We employ definitions from design theory consistent with [2]. An orthogonal array OA (k; n) is a k by n 2 array A whose entries come from an n-element set X, so that for any 1 i 1 ! i 2 k, and any ff; fi 2 X, there are exactly columns in the set fj : A[i 1 ; j] = ff and A[i 2 ; j] = fig. The orthogonal array is said to have order n, degree k and index . A (s; v; )--difference matrix over the group (G; ) of order s is a v by s matrix D with entries from G such that the multiset fD[i 1 ; j] D[i 2 ; j] \Gamma1 : j = 1; 2; : : : sg for any pair of rows (i 1 ; i 2 ) contains every element of G exactly times. If the base group G is abelian, additive notation..