On the Existence of de Bruijn Tori with Two by Two Windows

Abstract

Necessary and sufficient conditions for the existence of de Bruijn Tori (or Perfect Maps) with two by two windows over any alphabet are given. This is the first twodimensional window size for which the existence question has been completely answered for every alphabet. The techniques used to construct these arrays utilise existing results on Perfect Factors and Perfect Multi-Factors in one and two dimensions and involve new results on Perfect Factors with`puncturing capabilities'. Finally, the existence question for two-dimensional Perfect Factors is considered and is settled for two by two windows and alphabets of prime-power size. 1996 Academic Press, Inc.

Perfect Codes and Related Structures

“…There exists a PF(n, 5) with sequences of odd weight for 6 ≤ n ≤ 14 and n ∈ {16, 21, 22, 24, 28, 31}. For n ∈ {15, 17,18,19,20,23,25,26,27, 29, 30} we do not know whether such a perfect factor exists.…”

“…The are of de Bruijn arrays (which were generally called perfect maps) were extensively covered in the literature. Some more work on this topic and related structures was done by [23]- [25], [30], [32]- [37] The goal of the current paper is to extend the results on de Bruijn arrays into de Bruijn arrays codes defined as follows.…”

Growing perfect cubes

“…One dimensional perfect arrays are often called de Bruijn [4] or Good [7] sequences. Two dimensional perfect arrays are called also perfect maps [16] or de Bruijn tori [8][9][10].…”