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.
Similar content being viewed by others
References
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].
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.
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.
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.
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.
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.
Phelps, K.T., A General Product Construction for Error-Correcting Codes, SIAM J. Algebr. Discrete Methods, 1984, vol. 5, no. 2, pp. 224–228.
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.
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.
Phelps, K.T., An Enumeration of 1-Perfect Binary Codes of Length 15, Australas. J. Combin., 2000, vol. 21, pp. 287–298.
Vasil’ev, Yu.L., On Nongroup Closely Packed Codes, Probl. Kibern., 1962, vol. 8, pp. 337–339.
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.
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].
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1134/S0032946007040047