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Decoding algorithms of monotone codes and azinv codes and their unified view

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Abstract

This paper investigates linear-time decoding algorithms for two classes of error-correcting codes. One of the classes is monotone codes which are known as single deletion error-correcting codes, although they are not known to be single substitution error-correcting codes. The other is azinv codes which are known as single balanced adjacent deletion error-correcting codes, although they are not known to be single balanced adjacent substitution error-correcting codes. As a result, this paper proposes generalizations of Levenshtein’s decoding algorithm for Levenshtein’s single deletion or single substitution error-correcting codes. This paper points out that it is possible to unify our new two decoding algorithms. Moreover, we provide Python implementations of these algorithms and the graphs of their computational costs at https://github.com/Hokuto496/Decoding_Algorithms_of_monotone_codes_and_azinv_codes.

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References

  1. Buschmann T., Bystrykh L.V.: Levenshtein error-correcting barcodes for multiplexed DNA sequencing. BMC Bioinform. 14(1), 272 (2013).

    Article  Google Scholar 

  2. Chee Y.M., Kiah H.M., Vardy A., Yaakobi E., et al.: Coding for racetrack memories. IEEE Trans. Inform. Theory 64(11), 7094–7112 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  3. Gabrys R., Yaakobi E., Milenkovic O.: Codes in the Damerau distance for deletion and adjacent transposition correction. IEEE Trans. Inform. Theory 64(4), 2550–2570 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  4. Hagiwara M.: On ordered syndromes for multi insertion/deletion error-correcting codes. In: 2016 IEEE international symposium on information theory (ISIT), pp. 625–629. IEEE (2016)

  5. Hagiwara M.: Perfect codes for single balanced adjacent deletions. In: 2017 IEEE international symposium on information theory (ISIT), pp. 1938–1942. IEEE (2017)

  6. Helberg A.S., Ferreira H.C.: On multiple insertion/deletion correcting codes. IEEE Trans. Inform. Theory 48(1), 305–308 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  7. Inoue M., Kaneko H.: Deletion/insertion/reversal error correcting codes for bit-patterned media recording. In: 2011 IEEE international symposium on defect and fault tolerance in VLSI and nanotechnology systems, pp. 286–293. IEEE (2011)

  8. Krishnan A.R., Vasic B.: Coding for correcting insertions and deletions in bit-patterned media recording. In: 2011 IEEE global telecommunications conference-GLOBECOM 2011, pp. 1–5. IEEE (2011)

  9. Levenshtein V.: Binary codes capable of correcting deletions, insertions, and reversals. Sov. Phys. Dokl. 10(8), 707–710 (1966).

    MathSciNet  Google Scholar 

  10. Levenshtein V.: Asymptotically optimum binary code with correction for losses of one or two adjacent bits. Problemy Kibernet 19, 293–298 (1967).

    Google Scholar 

  11. Nakada K.: A family of perfectness of the levenshtein codes l a (n; 2n). In: 2020 international symposium on information theory and its applications (ISITA), pp. 131–135. IEEE (2020)

  12. Sima J., Bruck J.: Correcting deletions in multiple-heads racetrack memories. In: 2019 IEEE international symposium on information theory (ISIT), pp. 1367–1371. IEEE (2019)

  13. Song W., Polyanskii N., Cai K., He X.: Systematic single-deletion multiple-substitution correcting codes. arXiv preprint arXiv:2006.11516 (2020)

  14. Song W., Polyanskii N., Cai K., He X.: On multiple-deletion multiple-substitution correcting codes. In: 2021 IEEE international symposium on information theory (ISIT), pp. 2655–2660 (2021). https://doi.org/10.1109/ISIT45174.2021.9517878

  15. Takahashi H., Hagiwara M.: On generalizations of Levenshtein’s decoding algorithms (in japanese). In: Proceedings of SITA2019, pp. 13–18 (2019)

  16. Takahashi H., Hagiwara M.: Decoding algorithms of monotone codes and azinv codes and their unified view. In: 2020 international symposium on information theory and its applications (ISITA), pp. 284–288. IEEE (2020)

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Acknowledgements

This paper is partially supported by KAKENHI 18H01435.

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Authors and Affiliations

  1. Graduate School of Science and Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba-city, Chiba, Japan

    Hokuto Takahashi

  2. Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba-city, Chiba, Japan

    Manabu Hagiwara

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  1. Hokuto Takahashi
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  2. Manabu Hagiwara
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Correspondence to Hokuto Takahashi.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless"

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Takahashi, H., Hagiwara, M. Decoding algorithms of monotone codes and azinv codes and their unified view. Des. Codes Cryptogr. 90, 2893–2922 (2022). https://doi.org/10.1007/s10623-021-01004-0

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  • DOI: https://doi.org/10.1007/s10623-021-01004-0

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