A MESHING TECHNIQUE FOR de BRUIJN TORI

An (R, S; m, n)k-de Bruijn torus is a k-ary R × S toroidal array with the property that every k-ary m × n matrix appears exactly once contiguously on the torus. The torus is a generalization of de Bruijn cycles and has been extended to higher dimensions by many authors. The central question, asked by Chung, Diaconis, and Graham, is for which R, S, m, n, and k such tori exist. In this note we develop a notion of equivalence class de Bruijn cycles and we extend a technique of Iványi and Tóth. Combining these ideas we are able to construct the first examples in which R and S are not powers of k. We prove for all natural numbers s and t there is a (4st, 4st; 2, 2)2st-de Bruijn torus.