Abstract
A covering code, or a covering, is a set of codewords such that the union of balls centered at these codewords covers the entire space. As a rule, the problem consists in finding the minimum cardinality of a covering code. For the classical Hamming metric, the size of the smallest covering code of a fixed radius \(R\) is known up to a constant factor. A similar result has recently been obtained for codes with \(R\) insertions and for codes with \(R\) deletions. In the present paper we study coverings of a space for the fixed length Levenshtein metric, i.e., for \(R\) insertions and \(R\) deletions. For \(R=1\) and \(2\), we prove new lower and upper bounds on the minimum cardinality of a covering code, which differ by a constant factor only.
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References
Cohen, G., Honkala, I., Listyn, S., and Lobstein, A., Covering Codes, Amsterdam: Elsevier, 1997.
Smolensky, R., On Representations by Low-Degree Polynomials, in Proc. 34th Annu. Symp. on Foundations of Computer Science, Palo Alto, CA, USA, Nov. 3–5, 1993, pp. 130–138. https://doi.org/10.1109/SFCS.1993.366874
Pagh, R., Locality-sensitive Hashing without False Negatives, in Proc. 27th Annu. ACM–SIAM Symp. on Discrete Algorithms (SODA’2016), Arlington, VA, USA, Jan. 10–12, 2016, pp. 1–9. https://doi.org/10.1137/1.9781611974331.ch1
Micciancio, D., Almost Perfect Lattices, the Covering Radius Problem, and Applications to Ajtai’s Connection Factor, SIAM J. Comput., 2004, vol. 34, no. 1, pp. 118–169. https://doi.org/10.1137/S0097539703433511
Hämäläinen, H., Honkala, I., Litsyn, S., and Östergård, P., Football Pools—A Game for Mathematicians, Amer. Math. Monthly, 1995, vol. 102, no. 7, pp. 579–588. https://doi.org/10.2307/2974552
Ceze, L., Nivala, J., and Strauss, K., Molecular Digital Data Storage Using DNA, Nat. Rev. Genet., 2019, vol. 20, no. 8, pp. 456–466. https://doi.org/10.1038/s41576-019-0125-3
Bornholt, J., Lopez, R., Carmean, D.M., Ceze, L., Seelig, G., and Strauss, K., A DNA-Based Archival Storage System, in Proc. 21st Int. Conf. on Architectural Support for Programming Languages and Operating Systems (ASPLOS’16), Atlanta, GA, USA, Apr. 2–6, 2016, pp. 637–649. https://doi.org/10.1145/2872362.2872397
Church, G.M., Gao, Y., and Kosuri, S., Next-Generation Digital Information Storage in DNA, Science, 2012, vol. 337, no. 6102, p. 1628. https://doi.org/10.1126/science.1226355
Kabatiansky, G.A. and Panchenko, V.I., Packings and Coverings of the Hamming Space by Unit Spheres, Dokl. Akad. Nauk SSSR, 1988, vol. 303, no. 3, pp. 550–552 [Soviet Math. Dokl. (Engl. Transl.), 1989, vol. 38, no. 3, pp. 564–566]. https://www.mathnet.ru/eng/dan7368
Krivelevich, M., Sudakov, B., and Vu, V.H., Covering Codes with Improved Density, IEEE Trans. Inform. Theory, 2003, vol. 49, no. 7, pp. 1812–1815. https://doi.org/10.1109/TIT.2003.813490
Lenz, A., Rashtchian, C., Siegel, P.H., and Yaakobi, E., Covering Codes Using Insertions or Deletions, IEEE Trans. Inform. Theory, 2020, vol. 67, no. 6, pp. 3376–3388. https://doi.org/10.1109/TIT.2020.2985691
Fazeli, A., Vardy, A., and Yaakobi, E., Generalized Sphere Packing Bound, IEEE Trans. Inform. Theory, 2015, vol. 61, no. 5, pp. 2313–2334. https://doi.org/10.1109/TIT.2015.2413418
Applegate, D., Rains, E.M., and Sloane, N.J.A., On Asymmetric Coverings and Covering Numbers, J. Combin. Des., 2003, vol. 11, no. 3, pp. 218–228. https://doi.org/10.1002/jcd.10022
Sala, F. and Dolecek, L., Counting Sequences Obtained from the Synchronization Channel, in Proc. 2013 IEEE Int. Symp. on Information Theory (ISIT’2013), Istanbul, Turkey, July 7–12, 2013, pp. 2925–2929. https://doi.org/10.1109/ISIT.2013.6620761
Hoeffding, W., Probability Inequalities for Sums of Bounded Random Variables, J. Amer. Statist. Assoc., 1963, vol. 58, no. 301, pp. 13–30. https://doi.org/10.2307/2282952. Reprinted in: The Collected Works of Wassily Hoeffding, Fisher, N.I. and Sen, P.K., Eds., New York: Springer, 1994, pp. 409–426.
Wang, G. and Wang, Q., On the Size Distribution of Levenshtein Balls with Radius One, arXiv:2204.02201 [cs.IT], 2022.
He, L. and Ye, M., The Size of Levenshtein Ball with Radius 2: Expectation and Concentration Bound, in Proc. 2023 IEEE Int. Symp. on Information Theory (ISIT’2023), Taipei, Taiwan, June 25–30, 2023, pp. 850–855. https://doi.org/10.1109/ISIT54713.2023.10206888
Cooper, J.N., Ellis, R.B., and Kahng, A.B., Asymmetric Binary Covering Codes, J. Combin. Theory Ser. A, 2002, vol. 100, no. 2, pp. 232–249. https://doi.org/10.1006/jcta.2002.3290
Levenshtein, V.I., Binary Codes Capable of Correcting Deletions, Insertions, and Reversals, Dokl. Akad. Nauk SSSR, 1965, vol. 163, no. 4, pp. 845–848 [Soviet Phys. Dokl. (Engl. Transl.), 1966, vol. 10, no. 8, pp. 707–710]. http://mi.mathnet.ru/eng/dan31411
Levenshtein, V.I., Efficient Reconstruction of Sequences from Their Subsequences or Supersequences, J. Combin. Theory Ser. A, 2001, vol. 93, no. 2, pp. 310–332. https://doi.org/10.1006/jcta.2000.3081
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The research was carried out at the expense of the Russian Science Foundation, project no. 22-41-02028.
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Translated from Problemy Peredachi Informatsii, 2023, Vol. 59, No. 2, pp. 18–31. https://doi.org/10.31857/S055529232302002X
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Vorobyev, I.V. Covering Codes for the Fixed Length Levenshtein Metric. Probl Inf Transm 59, 86–98 (2023). https://doi.org/10.1134/S0032946023020023
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DOI: https://doi.org/10.1134/S0032946023020023