Partition and the Perfect Codes in the Additive Channel

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 con nected 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.