Nested perfect toroidal arrays

TL;DR

This paper introduces nested perfect toroidal arrays, a novel variant of de Bruijn tori, with recursive occurrence properties, constructed using Pascal triangle matrices modulo 2, offering a large set of such arrays for powers of two.

Contribution

It presents a new recursive array construction method based on Pascal matrices, expanding the class of perfect toroidal arrays with specific partitioning properties.

Findings

Constructs $2^{n^2+n-1}$ arrays for $n$ a power of 2

Arrays partition positions into congruence classes with unique occurrences

Method generalizes de Bruijn tori to nested, recursive structures

Abstract

We introduce two-dimensional toroidal arrays that are a variant of the de Bruijn tori. We call them nested perfect toroidal arrays. Instead of asking that every array of a given size has exactly one occurrence, we partition the positions in congruence classes and we ask exactly one occurrence in each congruence class. We also ask that this property applies recursively to each of the subarrays. We give a method to construct nested perfect toroidal arrays based on Pascal triangle matrix modulo 2. For the two-symbol alphabet, and for $n$ being a power of $2$, our method yields $2^{n^2+n-1}$ different nested perfect toroidal arrays allocating all the different $n\times n$ arrays in each congruence class that arises from taking the line number modulo $n$ and the column number modulo $n$.