We establish that the logarithm of the number of latin d‐cubes of order n is Θ(ndlnn) and the logarithm of the number of sets of t (t≥2 is fixed) orthogonal latin squares of order n is Θ(n2lnn). Similar estimations are obtained for systems of mutually strongly orthogonal latin d‐cubes. As a consequence, we construct a set of Steiner quadruple systems of order n such that the logarithm of its cardinality is Θ(n3lnn) as n→∞ and n mod 6=2 or 4.
Journal of Combinatorial Designs – Wiley
Published: May 1, 2018
Keywords: ; ; ; ; ; ; ; ;
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