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Partition and the Perfect Codes in the Additive Channel

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DOI: 10.4236/ojdm.2013.33021    2,124 Downloads   3,950 Views   Citations
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ABSTRACT

Many problems of discrete optimization are connected with partition of the n-dimensional space into certain subsets, and the requirements needed for these subsets can be geometricalfor instance, their sphericityor they can be connected with certain metricsfor instance, the requirement that subsets are Dirichlets regions with Hamming’s metrics [1]. Often partitions into some subsets are considered, on which a functional is optimized [2]. In the present work, the partitions of the n-dimensional space into subsets with zero limitation are considered. Such partitions allow us to construct the set of the group codes, V, and the set of the channels, A, between the arbitrary elements, V and A, having correcting relation between them. Descriptions of some classes of both perfect and imperfect codes in the additive channel are presented, too. A way of constructing of group codes correcting the errors in the additive channels is presented, and this method is a further generalization of Hammings method of code construction.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

G. Movsisyan, "Partition and the Perfect Codes in the Additive Channel," Open Journal of Discrete Mathematics, Vol. 3 No. 3, 2013, pp. 112-122. doi: 10.4236/ojdm.2013.33021.

References

[1] V. K. Leontyev, G. L. Movsisyan and J. G. Margaryan, “Partition of N-Dimensional Space on GF(2) into Dirichlet’s Regions,” Vestnik RAU, Vol. 2, No. 1, 2011, pp. 26-41.
[2] G. L. Movsisyan and J. G. Margaryan, “Optimal Sets in the N-Dimensional Cube,” Uchenie Zapiski, Vol. 170, No. 1, 1989, pp. 18-26.
[3] V. K. Leontyev, G. L. Movsisyan and J. G. Margaryan, “Perfect Codes in Additive Channels,” Reports of RAS, Vol. 411, No. 3, 2006, pp. 306-309.
[4] V. K. Leontyev, G. L. Movsisyan and J. G. Margaryan, “Correction of Errors in the Additive Channel,” Vestnik RAU, Vol. 2, No. 1, 2010, pp. 12-25.
[5] V. K. Leontyev, G. L. Movsisyan and J. G. Margaryan, “On Perfect Codes in Additive Channels,” Problems of Information Communication, Vol. 44, No. 4, 2008, pp. 12-19.
[6] F. J. M. Williams and N. J. A. Sloane, “The Theory of Error-Correcting Codes,” Bell Laboratories, Marray Hill, 1977.
[7] N. A. Solovyov, “Tests (Theory, Construction, Use),” Science, Novosibirsk, 1978, p. 189.

  
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