Constructions of transitive latin hypercubes. (May 2016)

• Record Type: Journal Article
• Authors: Krotov, Denis S.; Potapov, Vladimir N.
• Is Part Of: European journal of combinatorics. Volume 54(2016:May)
• Page Start: 51
• Page End: 64
• Publication Date: 2016-05
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2015.12.001
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: A function f : { 0, …, q − 1 } n → { 0, …, q − 1 } invertible in each argument is called a latin hypercube. A collection ( π 0, π 1, …, π n ) of permutations of { 0, …, q − 1 } is called an autotopism of a latin hypercube f if π 0 f ( x 1, …, x n ) = f ( π 1 x 1, …, π 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 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 .