Difference families, skew Hadamard matrices, and critical groups of doubly regular tournaments

Venkata Raghu Tej Pantangi¹

¹University of Florida

The electronic journal of combinatorics, Tome 26 (2019) no. 3

Résumé

In this paper we investigate the structure of the critical groups of doubly-regular tournaments (DRTs) associated with skew Hadamard difference families (SDFs) with one, two, or four blocks. Brown and Ried found that the existence of a skew Hadamard matrix of order $n+1$ is equivalent to the existence of a DRT on $n$ vertices. A well known construction of a skew Hadamard matrix order $n$ is by constructing skew Hadamard difference sets in abelian groups of order $n-1$. The Paley skew Hadamard matrix is an example of one such construction. Szekeres and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with two blocks. Wallis and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with four blocks. In this paper we consider the critical groups of DRTs associated with skew Hadamard matrices constructed from skew Hadamard difference families with one, two or four blocks. We compute the critical groups of DRTs associated with skew Hadamard difference families with two or four blocks. We also compute the critical group of the Paley tournament and show that this tournament is inequivalent to the other DRTs we considered. Consequently we prove that the associated skew Hadamard matrices are not equivalent.

Détail

DOI : 10.37236/8753

Classification : 05B20, 05B10, 05C50, 05C25, 05C20, 15A21
Mots-clés : skew Hadamard difference families, Hadamard difference families