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Keywords = orthogonal de Bruijn sequences

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20 pages, 342 KiB  
Article
Generalized Orthogonal de Bruijn and Kautz Sequences
by Yuan-Pon Chen, Jin Sima and Olgica Milenkovic
Entropy 2025, 27(4), 366; https://doi.org/10.3390/e27040366 - 30 Mar 2025
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Abstract
A de Bruijn sequence of order k over a finite alphabet is a cyclic sequence with the property that it contains every possible k-sequence as a substring exactly once. Orthogonal de Bruijn sequences are the collections of de Bruijn sequences of the [...] Read more.
A de Bruijn sequence of order k over a finite alphabet is a cyclic sequence with the property that it contains every possible k-sequence as a substring exactly once. Orthogonal de Bruijn sequences are the collections of de Bruijn sequences of the same order, k, that satisfy the joint constraint that every (k+1)-sequence appears as a substring in, at most, one of the sequences in the collection. Both de Bruijn and orthogonal de Bruijn sequences have found numerous applications in synthetic biology, although the latter remain largely unexplored in the coding theory literature. Here, we study three relevant practical generalizations of orthogonal de Bruijn sequences, where we relax either the constraint that every (k+1)-sequence appears exactly once or the sequences themselves are de Bruijn rather than balanced de Bruijn sequences. We also provide lower and upper bounds on the number of fixed-weight orthogonal de Bruijn sequences. The paper concludes with parallel results for orthogonal nonbinary Kautz sequences, which satisfy similar constraints as de Bruijn sequences, except for being only required to cover all subsequences of length k whose maximum run length equals one. Full article
(This article belongs to the Special Issue Coding and Algorithms for DNA-Based Data Storage Systems)
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