Extending de Bruijn sequences to larger alphabets
Section snippets
Introduction and statement of results
A circular sequence is the equivalence class of a sequence under rotations. A de Bruijn sequence of order n over a k-symbol alphabet is a circular sequence of length in which every length-n sequence occurs exactly once [6], [11], see [4] for a fine presentation and history. For example, writing to denote the circular sequence formed by the rotations of abc, [0011] is de Bruijn of order 2 over the alphabet .
A subsequence of a sequence is a sequence defined by
Proof of Theorem 1
In the sequel we use the terms word and sequence interchangeably. A de Bruijn graph is a directed graph whose vertices are the words of length n over a k-symbol alphabet and whose edges are the pairs where and , for some word u of length and possibly two different symbols . Thus, the graph has vertices and edges, it is strongly connected and every vertex has the same in-degree and out-degree. Each de Bruijn sequence of order n over a k-symbol alphabet
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We are most grateful to one of the referees for their significant recommendations. This research was done under grant Universidad de Buenos Aires UBACyT 20020170100309BA.
References (13)
- et al.
Perfect necklaces
Adv. Appl. Math.
(2016) - et al.
On extending de Bruijn sequences
Inf. Process. Lett.
(2011) - et al.
Normal numbers and nested perfect necklaces
J. Complex.
(2019) - et al.
The origins of combinatorics on words
Eur. J. Comb.
(2007) - et al.
On semi-perfect de Bruijn words
Theor. Comput. Sci.
(2018) - et al.
Introduction to Algorithms
(2009)