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 OJDM  Vol.3 No.3 , July 2013
Partition and the Perfect Codes in the Additive Channel
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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 geometrical—for instance, their sphericity—or they can be connected with certain metrics—for instance, the requirement that subsets are Dirichlet’s 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 Hamming’s method of code construction.
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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