Abstract
We give a one-step construction of de Bruijn sequences of general alphabet size and with order \(n+k\), given a de Bruijn sequence of order n and any integer \(k>1\). This is achieved by using an appropriate class of graph homomorphisms between de Bruijn digraphs whose orders differ by an integer k. The method starts with a lower order de Bruijn cycle, finds its inverse cycles in the higher order digraph, which are then cross-joined into one full cycle. Therefore, this generalizes the Lempel’s binary construction and the Alhakim–Akinwande construction for non-binary alphabets and a wide class of homomorphisms.
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We would like to thank the referees for their careful review and sharp remarks that helped us greatly improved the quality and the presentation of the paper.
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Communicated by L. Teirlinck.
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Alhakim, A., Nouiehed, M. Stretching de Bruijn sequences. Des. Codes Cryptogr. 85, 381–394 (2017). https://doi.org/10.1007/s10623-016-0314-4
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DOI: https://doi.org/10.1007/s10623-016-0314-4
Keywords
- de Bruijn sequence
- de Bruijn graph homomorphism
- Lempel’s D-morphism
- Recursive construction
- Linear feedback shift register