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Binary de Bruijn Sequences via Zech’s Logarithms

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Abstract

The focus of this work is to show how to combine Zech’s logarithms and each of the cycle joining and cross-join pairing methods to construct binary de Bruijn sequences. Basic implementations are supplied as proofs of concept. The cycles, in the cycle joining method, are typically generated by a linear feedback shift register. We prove a crucial characterization that determining Zech’s logarithms is equivalent to identifying conjugate pairs shared by any two distinct cycles. This speeds up the task of building a connected adjacency subgraph that contains all vertices of the complete adjacency graph. Distinct spanning trees in either graph correspond to cyclically inequivalent de Bruijn sequences. As the cycles are being joined, guided by the conjugate pairs, we track the changes in the feedback function. We show how to produce certificates of existence for spanning trees of certain types to conveniently handle large order cases. The characterization of conjugate pairs via Zech’s logarithms, as positional markings, is then adapted to identify cross-join pairs. A modified m-sequence is initially used, for ease of generation. The process can be repeated on each of the resulting de Bruijn sequences. Most prior constructions in the literature measure the complexity of the corresponding bit-by-bit algorithms. Our approach is different. We aim first to build a connected adjacency subgraph that is certified to contain all of the cycles as vertices. The ingredients are computed just once and concisely stored. Simple strategies are offered to keep the complexities low as the order grows.

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Notes

  1. The tables for characteristic 2 are included in the current standard distribution as well as in the online calculator for \(n \le 300\).

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Acknowledgements

M. F. Ezerman gratefully acknowledges the hospitality of the School of Mathematics and Statistics of Zhengzhou University, in particular Prof. Yongcheng Zhao, during several visits. The work of Z. Chang is supported by the National Natural Science Foundation of China Grant 61772476 and the Key Scientific Research Projects of Colleges and Universities in Henan Province Grant 18A110029. Grants TL-9014101684-01 and MOE2013-T2-1-041 support the research carried out by M. F. Ezerman, S. Ling, and H. Wang. Singapore Ministry of Education Grant M4011381 provides a partial support for A. A. Fahreza. J. Szmidt’s work is funded by the Polish Ministry of Science and Higher Education, Decision no. 6663/E-294/S/2018.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Zuling Chang

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore

    Martianus Frederic Ezerman, Adamas Aqsa Fahreza, San Ling & Huaxiong Wang

  3. Military Communication Institute, ul. Warszawska 22 A, 05-130, Zegrze, Poland

    Janusz Szmidt

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  1. Zuling Chang
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  2. Martianus Frederic Ezerman
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  3. Adamas Aqsa Fahreza
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  5. Janusz Szmidt
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Correspondence to Martianus Frederic Ezerman.

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Chang, Z., Ezerman, M.F., Fahreza, A.A. et al. Binary de Bruijn Sequences via Zech’s Logarithms. SN COMPUT. SCI. 2, 314 (2021). https://doi.org/10.1007/s42979-021-00683-9

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