On the de Bruijn Torus problem

Abstract

A ( k n ; n) k -de Bruijn Cycle is a cyclic k-ary sequence with the property that every k-ary n-tuple appears exactly once contiguously on the cycle. A ( k r , k s ; m, n) k -de Bruijn Torus is a k-ary k r × k s toroidal array with the property that every k-ary m × n matrix appears exactly once contiguously on the torus. As is the case with de Bruijn cycles, the 2-dimensional version has many interesting applications, from coding and communications to pseudo-random arrays, spectral imaging, and robot self-location. J. C. Cock proved the existence of such tori for all m, n, and k, and Chung, Diaconis, and Graham asked if it were possible that r = s and m = n for n even. Fan, Fan, Ma, and Siu showed this was possible for k = 2. Combining new techniques with old, we prove the result for k ⩾ 2 and show that actually much more is possible. The cases in 3 or more dimensions remain.