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arXiv · 1510.06212

On the number of SQSs, latin hypercubes and MDS codes

Vladimir N. Potapov
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Abstract

It is established that the logarithm of the number of latin $d$-cubes of order $n$ is $Θ(n^{d}\ln n)$ and the logarithm of the number of pairs of orthogonal latin squares of order $n$ is $Θ(n^2\ln n)$. 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 $Θ(n^3\ln n)$ as $n\rightarrow\infty$ and $n\ {\rm mod}\ 6= 2\ {\rm or}\ 4$.

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Vladimir N. Potapov. 2017-08-04. On the number of SQSs, latin hypercubes and MDS codes. https://doi.org/10.1002/jcd.21603

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