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Perfect binary codes of infinite length

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Abstract

A subset C of infinite-dimensional binary cube is called a perfect binary code with distance 3 if all balls of radius 1 (in the Hamming metric) with centers in C are pairwise disjoint and their union cover this binary cube. Similarly, we can define a perfect binary code in zero layer, consisting of all vectors of infinite-dimensional binary cube having finite supports. In this article we prove that the cardinality of all cosets of perfect binary codes in zero layer is the cardinality of the continuum. Moreover, the cardinality of all cosets of perfect binary codes in the whole binary cube is equal to the cardinality of the hypercontinuum.

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References

  1. S. V. Avgustinovich and F. I. Solov’eva, “Construction of Perfect Binary Codes by Sequential Shifts of ˜a-Components,” Problemy Peredachi Informatsii 33 (3), 15–21 (1997) [Problems Inform. Transmission 33 (3), 202–207 (1997)].

    MathSciNet  MATH  Google Scholar 

  2. Yu. L. Vasil’ev, “On Nongroup Close-Packed Codes,” in Problems of Cybernetics, Vol. 8, Ed. by A. A. Lyapunov (Fizmatgiz, Moscow, 1962), pp. 337–339.

    Google Scholar 

  3. S. A. Malyugin, “On Enumeration of the Perfect Binary Codes of Length 15,” Diskretn. Anal. Issled. Oper. Ser. 2, 6 (2), 48–73 (1999) [Discrete Appl. Math. 135 (1–3), 161–181 (2004)].

    MathSciNet  MATH  Google Scholar 

  4. S. A. Malyugin, “Nonsystematic Perfect Binary Codes,” Diskretn. Anal. Issled. Oper. Ser. 1, 8 (1), 55–76 (2001).

    MathSciNet  MATH  Google Scholar 

  5. S. A. Malyugin, “Perfect Binary Codes of Infinite Length,” Prikl. Diskretn. Mat., Prilozh. No. 8, 117–120 (2015).

    Article  Google Scholar 

  6. V. N. Potapov, “Infinite-Dimensional Quasigroups of Finite Orders,” Mat. Zametki 93 (3), 457–465 (2013) [Math. Notes 93 (3), 479–486 (2013)].

    Article  MathSciNet  MATH  Google Scholar 

  7. A. M. Romanov, “On Construction of Perfect Nonlinear Binary Codes by Symbol Inversion,” Diskretn. Anal. Issled. Oper. Ser. 1, 4 (1), 46–52 (1997).

    MathSciNet  MATH  Google Scholar 

  8. K. T. Phelps and M. LeVan, “Kernels of Nonlinear Hamming Codes,” Des. Codes Cryptogr. 6 (3), 247–257 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  9. F. I. Solov’eva, “Switchings and Perfect Codes,” in Numbers, Information, and Complexity, Vol. I, Ed. by Althöfer et al. (Kluwer Acad. Publ., Dordrecht, 2000), pp. 311–324.

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    S. A. Malyugin

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  1. S. A. Malyugin
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Correspondence to S. A. Malyugin.

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Original Russian Text © S.A. Malyugin, 2017, published in Diskretnyi Analiz i Issledovanie Operatsii, 2017, Vol. 24, No. 2, pp. 53–67.

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Malyugin, S.A. Perfect binary codes of infinite length. J. Appl. Ind. Math. 11, 227–235 (2017). https://doi.org/10.1134/S1990478917020089

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  • DOI: https://doi.org/10.1134/S1990478917020089

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