Modified generalized Hadamard matrices and constructions for transversal designs

Abstract

It is well known that there exists a transversal design TD λ [k; u] admitting a class regular automorphism group U if and only if there exists a generalized Hadamard matrix GH(u, λ) over U . Note that in this case the resulting transversal design is symmetric by Jungnickel's result. In this article we define a modified generalized Hadamard matrix and show that transversal designs which are not necessarily symmetric can be constructed from these under a modified condition similar to class regularity even if it admits no class regular automorphism group.

A construction for modified generalized Hadamard matrices using QGH matrices

“…In [6] we introduced the notion of a modified generalized Hadamard matrix over a group. We first give a summary of the related results, which we will use in the later sections.…”

“…Transversal designs obtained from G H(s, u, λ) matrices are not always symmetric (see Example 5.3 of [6]) and do not always admit class regular automorphism groups even if they are symmetric (see [7]). The following gives a criterion for the resulting transversal design to be symmetric.…”

“…From a generalized Hadamard matrix we obtain a symmetric transversal design admitting G as a class regular automorphism group [2]. On the other hand a modified generalized Hadamard matrix GH(s, u, λ) over a group is defined in [6] and from this one can construct a transversal design TD λ (uλ, u) admitting G as an automorphism group (see Result 2.2).…”

A family of non class-regular symmetric transversal designs of spread type

“…From each spread we obtain many GH(q, q, q) matrices [2] and we say that the resulting transversal designs are of spread type. …”

“…Result 1.1[2] Let [D i j ] be a t ×tGH(s, u, λ) matrix over a group H of order su with respect to subgroups U i (1 ≤ i ≤ t), where t = uλ/s. If we define P and B by…”