Constructions of transitive latin hypercubes

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 growths 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.

q-ary Propelinear Perfect Codes from the Regular Subgroups of the GA(r,q) and Their Ranks

“…Unlike binary codes, which are well-studied, there are rather few works devoted to q-ary propelinear codes, for q ≥ 3. We refer to [7], [2], [8], [1] for q-ary perfect, MDS and generalized Hadamard propelinear codes.…”