On the number of transversal designs

Bounds are obtained on the number of distinct transversal designs 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 ⩽ q + 1 where q is a prime power, then the number of nonisomorphic TD ( g , q r ) designs is at least q α r q 2 r ( 1 − o ( 1 ) ) as r → ∞ , where α = 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.