Constructions of transitive latin hypercubes

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 $\pi_0f(x_1,...,x_n)=f(\pi_1x_1,...,\pi_n x_n)$ 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 $q=5$. Keywords: transitive code, propelinear code, latin square, latin hypercube, autotopism, G-loop.Comment: 18 pages. v3: revised, accepted version; v2: the paper has been completely rewritten (v1 can contain incorrect statements