On the number of SQSs, latin hypercubes and MDS codes

Abstract

It is established that the logarithm of the number of latin $d$-cubes of order $n$ is $\Theta(n^{d}\ln n)$ and the logarithm of the number of pairs of orthogonal latin squares of order $n$ is $\Theta(n^2\ln n)$. Similar estimations are obtained for systems of mutually strong orthogonal latin $d$-cubes. As a consequence, it is constructed a set of Steiner quadruple systems of order $n$ such that the logarithm of its cardinality is $\Theta(n^3\ln n)$ as $n\rightarrow\infty$ and $n\ {\rm mod}\ 6= 2\ {\rm or}\ 4$.