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Updated: May 10, 2025

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Generalized Orthogonal de Bruijn and Kautz Sequences.

Yuan-Pon Chen1, Jin Sima1, Olgica Milenkovic1

  • 1Department of Electrical and Computer Engineering, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA.

Entropy (Basel, Switzerland)
|April 26, 2025
PubMed
Summary

This study explores generalized orthogonal de Bruijn sequences, crucial for synthetic biology. We introduce new sequence types and provide bounds for fixed-weight sequences, expanding coding theory applications.

Keywords:
De Bruijn sequencesKautz sequencesbalanced de Bruijn sequencesfixed-weight de Bruijn sequencesorthogonal Kautz sequencesorthogonal de Bruijn sequencessynthetic biology

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Area of Science:

  • Coding Theory
  • Combinatorics
  • Synthetic Biology

Background:

  • De Bruijn sequences are cyclic sequences containing all k-sequences exactly once.
  • Orthogonal de Bruijn sequences ensure unique (k+1)-sequences across a collection.
  • These sequences have applications in synthetic biology but are understudied in coding theory.

Purpose of the Study:

  • To investigate practical generalizations of orthogonal de Bruijn sequences.
  • To relax constraints on (k+1)-sequence occurrences or use balanced de Bruijn sequences.
  • To establish bounds for fixed-weight orthogonal de Bruijn sequences.

Main Methods:

  • Studying three generalizations of orthogonal de Bruijn sequences.
  • Analyzing relaxed constraints for (k+1)-sequence occurrences.
  • Deriving lower and upper bounds for fixed-weight orthogonal de Bruijn sequences.

Main Results:

  • Introduced and analyzed three practical generalizations of orthogonal de Bruijn sequences.
  • Provided bounds for the number of fixed-weight orthogonal de Bruijn sequences.
  • Extended results to orthogonal nonbinary Kautz sequences.

Conclusions:

  • The study expands the understanding and applicability of orthogonal de Bruijn sequences.
  • New sequence generalizations and bounds offer valuable tools for coding theory and synthetic biology.
  • Parallel results for Kautz sequences broaden the scope of combinatorial sequence applications.