On binary de Bruijn sequences from LFSRs with arbitrary characteristic polynomials

  • Zuling Chang
  • Martianus Frederic Ezerman
  • San Ling
  • Huaxiong Wang


We propose a construction of de Bruijn sequences by the cycle joining method from linear feedback shift registers (LFSRs) with arbitrary characteristic polynomial f(x). We study in detail the cycle structure of the set \(\varOmega (f(x))\) that contains all sequences produced by a specific LFSR on distinct inputs and provide a fast way to find a state of each cycle. This leads to an efficient algorithm to find all conjugate pairs between any two cycles, yielding the adjacency graph. The approach is practical to generate a large class of de Bruijn sequences up to order \(n \approx 20\). Many previously proposed constructions of de Bruijn sequences are shown to be special cases of our construction.


Binary periodic sequence LFSR de Bruijn sequence Cycle structure Adjacency graph Cyclotomic number 

Mathematics Subject Classification

11B50 94A55 94A60 



Adamas Aqsa Fahreza wrote the python implementation code. The work of Z. Chang is supported by the National Natural Science Foundation of China under Grant 61772476 and the Key Scientific Research Projects of Colleges and Universities in Henan Province under Grant 18A110029. Research Grants TL-9014101684-01 and MOE2013-T2-1-041 support the research carried out by M. F. Ezerman, S. Ling, and H. Wang. The authors gratefully acknowledge the advise and feedbacks from the editor and the reviewers. They led us to a better presentation of the results.


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouChina
  2. 2.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

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