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On partitions of an n-cube into nonequivalent perfect codes

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Abstract

We prove that for all n = 2k − 1, k ≥ 5, there exists a partition of the set of all binary vectors of length n into pairwise nonequivalent perfect binary codes of length n with distance 3.

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References

  1. Avgustinovich, S.V. and Solov’eva, F.I., Construction of Perfect Binary Codes by Sequential Shifts of ᾶ-Components, Probl. Peredachi Inf., 1997, vol. 33, no. 3, pp. 15–21 [Probl. Inf. Trans. (Engl. Transl.), 1997, vol. 33, no. 3, pp. 202–207].

    MathSciNet  Google Scholar 

  2. Solov’eva, F.I., On Binary Nongroup Codes, in Metody diskretnogo analiza v izuchenii bulevykh funktsii i grafov (Methods of Discrete Analysis in Studying Boolean Functions and Graphs), Novosibirsk: Inst. Mat. Sib. Otd. Akad. Nauk SSSR, 1981, vol. 37, pp. 65–76.

    Google Scholar 

  3. Rifà, J. and Vardy, A., On Partitions of Space into Perfect Codes, i Proc. 3rd Mediterranean Workshop on Coding and Information Integrity, Ein Boqeq, Israel, 1997.

  4. Borges, J., Fernández, C., Rifà, J., and Villanueva, M., Constructions of 1-Perfect Partitions on the n-Cube (Z/2)n, Tech. Report PIRDI 1/01, Escola Tècnica Superior d’Enginyeries (ETSE), Spain, 2001.

  5. Rifà, J., Well-ordered Steiner Triple Systems and 1-Perfect Partitons of the n-Cube, SIAM J. Discrete Math., 1999, vol. 12, no. 1, P. 35–47.

    Article  MathSciNet  MATH  Google Scholar 

  6. Rifà, J., Pujol, J., and Borges, J., 1-Perfect Uniform and Distance Invariant Partitions, Appl. Algebra Engrg. Comm. Comput., 2001, vol. 11, no. 4, pp. 297–311.

    Article  MathSciNet  MATH  Google Scholar 

  7. Phelps, K.T., A General Product Construction for Error-Correcting Codes, SIAM J. Algebr. Discrete Methods, 1984, vol. 5, no. 2, pp. 224–228.

    Article  MathSciNet  MATH  Google Scholar 

  8. Etzion, T. and Vardy, A., On Perfect Codes and Tilings: Problems and Solutions, SIAM J. Discrete Math., 1998, vol. 11, no. 2, pp. 205–223.

    Article  MathSciNet  MATH  Google Scholar 

  9. Avgustinovich, S.V., Lobstein, A.C., and Solov’eva, F.I., Intersection Matrices for Partitions by Binary Perfect Codes, IEEE Trans. Inform. Theory, 2001, vol. 47, no. 4, pp. 1621–1624.

    Article  MathSciNet  MATH  Google Scholar 

  10. Phelps, K.T., An Enumeration of 1-Perfect Binary Codes of Length 15, Australas. J. Combin., 2000, vol. 21, pp. 287–298.

    MathSciNet  MATH  Google Scholar 

  11. Vasil’ev, Yu.L., On Nongroup Closely Packed Codes, Probl. Kibern., 1962, vol. 8, pp. 337–339.

    Google Scholar 

  12. Solov’eva, F.I., On Perfect Codes and Related Topics, Com2Mac Lect. Note Ser., vol. 13, Pohang, Korea: Pohang Univ. of Science and Technology, 2004.

    Google Scholar 

  13. Avgustinovich, S.V. and Solov’eva, F.I., On the Nonsystematic Perfect Binary Codes, Probl. Peredachi Inf., 1996, vol. 32, no. 3, pp. 47–50 [Probl. Inf. Trans. (Engl. Transl.), 1996, vol. 32, no. 3, pp. 258–261].

    MathSciNet  Google Scholar 

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Correspondence to S. V. Avgustinovich.

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Original Russian Text © S.V. Avgustinovich, F.I. Solov’eva, O. Heden, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 4, pp. 45–50.

Supported in part by the Royal Swedish Academy of Sciences.

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Avgustinovich, S.V., Solov’eva, F.I. & Heden, O. On partitions of an n-cube into nonequivalent perfect codes. Probl Inf Transm 43, 310–315 (2007). https://doi.org/10.1134/S0032946007040047

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  • DOI: https://doi.org/10.1134/S0032946007040047

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