On the number of SQSs, latin hypercubes and MDS codes.

Abstract: We establish that the logarithm of the number of latin <italic>d</italic>‐cubes of order <italic>n</italic> is Θ ( n d ln n ) and the logarithm of the number of sets of <italic>t</italic> ( t ≥ 2 is fixed) orthogonal latin squares of order <italic>n</italic> is Θ ( n 2 ln n ). Similar estimations are obtained for systems of mutually strongly orthogonal latin <italic>d</italic>‐cubes. As a consequence, we construct a set of Steiner quadruple systems of order <italic>n</italic> such that the logarithm of its cardinality is Θ ( n 3 ln n ) as n → ∞ and n mod 6 = 2 or 4. [ABSTRACT FROM AUTHOR]