TITLE:
Partition and the Perfect Codes in the Additive Channel
AUTHORS:
Garib Movsisyan
KEYWORDS:
Partition; Perfect Codes; Additive Channel
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.3 No.3,
July
2,
2013
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
withcertain 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.