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 . Often partitions into some subsets are
considered, on which a functional is optimized .In the present work, the
partitions of the n-dimensional space
into subsets with “zero” limitation are considered. Such partitions allowus 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.
 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.
 G. L. Movsisyan and J. G. Margaryan, “Optimal Sets in the N-Dimensional Cube,” Uchenie Zapiski, Vol. 170, No. 1, 1989, pp. 18-26.
 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.
 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.
 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.
 F. J. M. Williams and N. J. A. Sloane, “The Theory of Error-Correcting Codes,” Bell Laboratories, Marray Hill, 1977.
 N. A. Solovyov, “Tests (Theory, Construction, Use),” Science, Novosibirsk, 1978, p. 189.