New Constructions for De Bruijn Tori

A De Bruijn torus is a periodic d−dimensional k−ary array such that each n1 × · · · × nd k−ary array appears exactly once with the same period. We describe two new methods of constructing such arrays. The first is a type of product that constructs a k1k2−ary torus from a k1−ary torus and a k2−ary torus. The second uses a decomposition of a d-dimensional torus to produce a d + 1 dimensional torus. Both constructions will produce two dimensional k−ary tori for which the period is not a power of k. In particular, for k = Πpl l and for all natural numbers (n1, n2), we construct 2-dimensional k−ary De Bruijn tori with order 〈n1, n2〉 and period 〈q, k 12/q〉 where q = kΠp pl n1c l .