Constructions of transitive latin hypercubes

A set of vertices of the q-ary n-dimensional Hamming space is called a distance-2 MDS code if it has exactly one element in intersection with every line. By an isotopism we mean an isometric transformation of the Hamming space that independently permutes the symbols in every coordinate but does not change the coordinate order. A set M of vertices is called transitive if its automorphism group (i.e., the stabilizer of M in the space isometry group) acts transitively on M . A set M is called isotopically transitive if its autotopy group acts transitively onM . We call a setM topolinear (propelinear) if it can be equipped with a group operation whose left multiplication can be always extended to an isotopism (respectively, a space isometry). It is proved that if the arity is even > 2 or divisible by a square of prime, then there exists a quasi-exponential (more tightly, exponential with respect to the square root of the space dimension) number of topolinear and, as a corollary, propelinear and isotopically transitive MDS codes. The set of isotopically transitive MDS codes in the 4-ary Hamming space is characterized.