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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.

References

  1. Bruckstein Alfred M., et al.: Simple and robust binary self-location patterns. IEEE Trans. Info. Theory 58(7), 4884–4889 (2012). https://doi.org/10.1109/TIT.2012.2191699.

    Article  MathSciNet  Google Scholar 

  2. Burns J., Mitchell C.J.: Coding schemes for two-dimensional position sensing. Technical Publications Department, Hewlett-Packard Laboratories (1992).

    Google Scholar 

  3. Chartrand G., Lesniak L., Zhang P.: Graphs & digraphs, vol. 39. CRC Press, Boca Raton (2010).

    Book  Google Scholar 

  4. Cock J.C.: Toroidal tilings from de Bruijn-Good cyclic sequences. Discret. Math. 70, 209–210 (1988).

    Article  MathSciNet  Google Scholar 

  5. de Bruijn N.G.: A combinatorial problem. Indagationes Math 49(1), 758–764 (1946).

    Google Scholar 

  6. Etzion T.: Constructions for perfect maps and pseudorandom arrays. IEEE Trans. Inf. Theory 34(5), 1308–1316 (1988).

    Article  MathSciNet  Google Scholar 

  7. Fan C.T., et al.: On de Bruijn arrays. Ars Combinatoria 19A, 205–213 (1985).

    MathSciNet  Google Scholar 

  8. Flye Sainte-Marie C.: Solution to problem number. L’Intermediare des Mathematiciens 1, 107–110 (1894).

    Google Scholar 

  9. Grubman T., Ahmet Şekercioğlu Y., Wood D.R.: Partitioning de Bruijn graphs into fixed-length cycles for robot identification and tracking. Discret. Appl. Math. 213, 101–113 (2016). https://doi.org/10.1016/j.dam.2016.05.013.

    Article  MathSciNet  Google Scholar 

  10. Horan V., Stevens B.: Locating patterns in the de Bruijn torus. Discret. Math. 339(4), 1274–1282 (2016).

    Article  MathSciNet  Google Scholar 

  11. Hurlbert G.H., Isaak G.: New constructions for De Bruijn tori. Des. Codes Cryptogr. 6, 47–56 (1995).

    Article  MathSciNet  Google Scholar 

  12. Paterson K.G.: New classes of perfect maps I. J. Comb. Theory Ser. A 73(2), 302–334 (1996).

    Article  MathSciNet  Google Scholar 

  13. Paterson K.G.: New classes of perfect maps II. J. Comb. Theory Ser. A 73(2), 335–345 (1996).

    Article  MathSciNet  Google Scholar 

  14. Pudwell L., Rockey R.: de Bruijn Arrays for L-Fillings. In: Mathematics Magazine 87(1), pp. 57–60 (2014). http://www.jstor.org/stable/10.4169/math.mag.87.1.57 (visited on 10/27/2022).

  15. Shibata Y., Hasunuma T., Fukuda S.: Isomorphic factorization of de Bruijn digraphs. Discret. Math. 218(1–3), 199–208 (2000).

    Article  MathSciNet  Google Scholar 

Download references

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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Authors and Affiliations

  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. 92, 1439–1454 (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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