Perfect factors in the de Bruijn graph

Abstract

Abstract.A Perfect Factor in the de Bruijn graph is a collection of disjoint cycles of a fixed period with the property that each vertex of the graph lies on exactly one cycle. Alternatively a Perfect Factor is a set of periodic sequences of a fixed period such that for some fixed u, every possible u-tuple of consecutive symbols occurs exactly once as a sub-sequence of a sequence of the set. As well as being of interest in their own right, Perfect Factors are fundamental in the construction of Perfect Maps in two dimensions. In this paper, necessary conditions on the period for the existence of Perfect Factors are given. These conditions are shown to be sufficient in the case where the size of the alphabet is a power of a prime. We also consider the construction of Perfect Factors over general c-ary alphabets. Our results are then applied to obtain a large new class of non-binary Perfect Maps.

Permutation Polynomials, de Bruijn Sequences, and Linear Complexity

“…Let d $= 16 shows that in order to prove that there exist permutations of F p m for every m 2 and for every degree, excluding those ruled out by our non-existence result, it is sufficient to prove the result for permutations over F p 2 . We have already noted that the case m=1 is equivalent to the existence of permutation polynomials of the appropriate degrees.…”

Constructions for higher dimensional perfect multifactors

“…Also, analogues of Corollaries 2.5 and 2.5 are needed. However, these can be obtained as in the proof of Corollary 2.5 using results from [17]. Lemma 3.2.…”

“…Likewise, if De Bruijn cycles were first discovered in [7] and later independently in [2] and [9] (see Frederickson [8] for a survey of De Bruijn cycles). 2-dimensional De Bruijn tori are examined in [1,[4][5][6][11][12][13][14][15][16][17][18], among others. A 2-dimensional De Bruijn torus (r 1 , r 2 ; n 1 , n 2 ; 1) 2 k is square if r 1 = r 2 = r and totally square if n 1 = n 2 = n as well.…”

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

“…The parameter sets satisfying Lemma 1 for which existence of de Bruijn tori remains unsettled after Result 2 are those where either p v for every i could be resolved if the existence question for one-dimensional perfect factors were positively settled, by using the generalisation of Etzion's construction given in [18]. Recent progress on this problem can be found in [15 17].…”

“…Square de Bruijn Tori (i.e., (R, R; u, u) C -dBT) were considered in [10], while some families of arrays with u=v=2 were constructed in [11]. The existence question for de Bruijn Tori over general alphabets was studied in [11,12,18,21], and higher dimensional versions were considered in [13].…”