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

Abstract

Many classes of symmetric transversal designs have been constructed from generalized Hadamard matrices and they are necessarily class regular. In (Hiramine, Des Codes Cryptogr 56:21-33, 2010) we constructed symmetric transversal designs using spreads of Z 2n p with p a prime. In this article we show that most of them admit no class regular automorphism groups. This implies that they are never obtained from generalized Hadamard matrices. As far as we know, this is the first infinite family of non class-regular symmetric transversal designs.

A construction for modified generalized Hadamard matrices using QGH matrices

“…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.…”