Constructions of transitive latin hypercubes Full article
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European Journal of Combinatorics
ISSN: 0195-6698 , E-ISSN: 1095-9971 |
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| Output data | Year: 2016, Volume: 54, Pages: 51-64 Pages count : 14 DOI: 10.1016/j.ejc.2015.12.001 | ||
| Tags | transitive code, propelinear code, latin square, latin hypercube, autotopism, G-loop | ||
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Abstract:
A function $f:\{0,...,q-1\}^n\to\{0,...,q-1\}$ invertible in each argument is called a latin hypercube. A collection $(\pi_0,\pi_1,...,\pi_n)$ of permutations of $\{0,...,q-1\}$ is called an autotopism of a latin hypercube $f$ if $\pi_0f(x_1,...,x_n)=f(\pi_1x_1,...,\pi_nx_n)$ for all $x_1$, ..., $x_n$. We call a latin hypercube isotopically transitive (topolinear) if its group of autotopisms acts transitively (regularly) on all $q^n$ collections of argument values. We prove that the number of nonequivalent topolinear latin hypercubes grows exponentially with respect to $\sqrt{n}$ if $q$ is even and exponentially with respect to $n^2$ if $q$ is divisible by a square. We show a connection of the class of isotopically transitive latin squares with the class of G-loops, known in noncommutative algebra, and establish the existence of a topolinear latin square that is not a group isotope. We characterize the class of isotopically transitive latin hypercubes of orders $q=4$ and $q=5$.
Cite:
Krotov D.S.
, Potapov V.N.
Constructions of transitive latin hypercubes
European Journal of Combinatorics. 2016. V.54. P.51-64. DOI: 10.1016/j.ejc.2015.12.001 WOS Scopus РИНЦ OpenAlex
Constructions of transitive latin hypercubes
European Journal of Combinatorics. 2016. V.54. P.51-64. DOI: 10.1016/j.ejc.2015.12.001 WOS Scopus РИНЦ OpenAlex
Dates:
| Submitted: | Apr 28, 2015 |
| Accepted: | Dec 2, 2015 |
| Published online: | Dec 24, 2015 |
Identifiers:
| Web of science: | WOS:000371360600004 |
| Scopus: | 2-s2.0-84950998135 |
| Elibrary: | 26801587 |
| OpenAlex: | W2113012511 |