On the number of SQSs, latin hypercubes and MDS codes

TL;DR

This paper provides asymptotic estimates for the number of Latin hypercubes, orthogonal Latin squares, and Steiner quadruple systems, revealing their growth rates as the order increases.

Contribution

It establishes new asymptotic bounds for the counts of Latin hypercubes, orthogonal Latin squares, and constructs large Steiner quadruple systems with specified properties.

Findings

Logarithm of Latin d-cubes count is Θ(n^d ln n)

Logarithm of orthogonal Latin squares pairs is Θ(n^2 ln n)

Constructed Steiner quadruple systems with size Θ(n^3 ln n)

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 $n\ {\rm mod}\ 6= 2\ {\rm or}\ 4$.