On the number of transversal designs

Donovan, D. M. and Grannell, M. J. (2013). On the number of transversal designs. Journal of Combinatorial Theory, Series A, 120(7) pp. 1562–1574.

DOI: https://doi.org/10.1016/j.jcta.2013.05.004

URL: http://www.sciencedirect.com/science/article/pii/S...

Abstract

Bounds are obtained on the number of distinct transversal designs TD $\TD(g,n)$ (having $g$ groups with $n$ points in each group) for certain values of $g$ and $n$ . Amongst other results it is proved that, if $2<g\le q+1$ where $q$ is a prime power, then the number of nonisomorphic TD $\TD(g,q^r)$ designs is at least $q^{\alpha rq^{2r}(1-o(1))}$ as $r\to\infty$ , where $\alpha=1/q^4$ . The bounds obtained give equivalent bounds for the numbers of distinct and nonisomorphic sets of $g-2$ mutually orthogonal Latin squares of order $n$ in the corresponding cases. Applications to other combinatorial designs are also described.