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Abstract:

A (kn;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 (kr, ks; m, n)k-de Bruijn Torus is a k-ary krXks toroidal array with the property that every k-ary m x 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.

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Or copy and paste:
   Glenn Hurlbert and Garth Isaak, "On The De Bruijn Torus Problem." Dartmouth Computer Science Technical Report PCS-TR92-181, 1992.

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