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 $n\to \infty$ and $n\mathrm{mod}6=2\mathrm{or}4$.

10 pages

References in corpus (1)

• On the Classification of MDS Codes

Cited by in corpus (3)

• Constructing MRD codes by switching
• Embedding in MDS codes and Latin cubes
• An upper bound on the number of frequency hypercubes