New constructions for De Bruijn tori

Abstract

A De Bruijn torus is a periodicd-dimensionalk-ary array such that eachn ₁ × ... ×n _d 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 ak ₁ k ₂ -ary torus from ak ₁ -ary torus and ak ₂ -ary torus. The second uses a decomposition of ad-dimensional torus to produce ad+1 dimensional torus. Both constructions will produce two dimensionalk-ary tori for which the period is not a power ofk. In particular, for$k = \Pi p_l^{\alpha _l }$ and for all natural numbers (n ₁ , n ₂ ), we construct 2-dimensionalk-ary De Bruijn tori with order 〈n ₁ , n ₂ 〉 and period$\langle q, k^{n_1 n_2 /q} \rangle$ where$q = k\Pi p_l^{\left\lfloor {\log _{p_l } n_1 } \right\rfloor }$.