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Using alternating de Bruijn sequences to construct de Bruijn tori

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Abstract

A de Bruijn torus is the two dimensional generalization of a de Bruijn sequence. While methods exist to generate these tori, only a few methods of construction are known. We present a novel method to generate de Bruijn tori with rectangular windows by combining two variants of de Bruijn sequences.

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Data availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

Notes

  1. The last row and column are adjacent to the first row and column respectively.

  2. \(\Pi \) being the rotations on strings of length r

  3. Here we use the variable convention in [4].

  4. i.e. we apply the cumulative rotations on each string.

  5. It is unknown how many de Bruijn families exist.

  6. Recall that i represents \(\pi _i\) etc.

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Acknowledgements

We would like to thank the University of Guelph for the opportunity to research these construction methods. We would also like to thank Daniel Ashlock for leading our research into this direction. Matthew Kreitzer was supported by the Ontario Graduate Scholarship (OGS) program. Mihai Nica was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery grant RGPIN-2021-02533. Rajesh Pereira was supported by the NSERC Discovery grant RGPIN-2022-04149.

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  1. University of Guelph, Guelph, Canada

    Matthew Kreitzer, Mihai Nica & Rajesh Pereira

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  1. Matthew Kreitzer
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  2. Mihai Nica
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  3. Rajesh Pereira
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Correspondence to Mihai Nica.

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Communicated by J. Jedwab.

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Kreitzer, M., Nica, M. & Pereira, R. Using alternating de Bruijn sequences to construct de Bruijn tori. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-023-01351-0

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  • DOI: https://doi.org/10.1007/s10623-023-01351-0

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