In the present work, the class of metrics connected with subsets of the linear space on the field, GF(2), is considered and a number of facts are established, which allow us to express the correcting capacity of codes...In the present work, the class of metrics connected with subsets of the linear space on the field, GF(2), is considered and a number of facts are established, which allow us to express the correcting capacity of codes for the additive channel in terms of this metrics. It is also considered a partition of the metric space, Bn, by means of D-representable codes. The equivalence of D-representable and the perfect codes in the additive channel is proved.展开更多
The problem of classification of the subset of the vertices of the n-dimensional unit cube in respect to all “shifts” by a vector from Bn is studied. Some applications for the investigation of the additive channels ...The problem of classification of the subset of the vertices of the n-dimensional unit cube in respect to all “shifts” by a vector from Bn is studied. Some applications for the investigation of the additive channels of communication are represented.展开更多
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...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.展开更多
In the present paper, geometry of the Boolean space B<sup>n</sup> in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressi...In the present paper, geometry of the Boolean space B<sup>n</sup> in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressions for representing of classical figures in B<sup>n</sup> and the functions of distances between them. In particular, equations in sets are considered and their interpretations in combinatory terms are given.展开更多
The notion of a communication channel is one of the key notions in information theory but like the notion “information” it has not any general mathematical definition. The existing examples of the communication chan...The notion of a communication channel is one of the key notions in information theory but like the notion “information” it has not any general mathematical definition. The existing examples of the communication channels: the Gaussian ones;the binary symmetric ones;the ones with symbol drop-out and drop-in;the ones with error packets etc., characterize the distortions which take place in information conducted through the corresponding channel.展开更多
In the present work, a construction making possible creation of an additive channel of cardinality s and rank r for arbitrary integers s, r, n (r≤min (n,s-1)), as well as creation of a code correcting err...In the present work, a construction making possible creation of an additive channel of cardinality s and rank r for arbitrary integers s, r, n (r≤min (n,s-1)), as well as creation of a code correcting errors of the channel A is presented.展开更多
In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operatio...In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space B<sup>n</sup> are presented. Also, sums and multisums of various “classical figures” as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation X+Y=A, where , are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.展开更多
文摘In the present work, the class of metrics connected with subsets of the linear space on the field, GF(2), is considered and a number of facts are established, which allow us to express the correcting capacity of codes for the additive channel in terms of this metrics. It is also considered a partition of the metric space, Bn, by means of D-representable codes. The equivalence of D-representable and the perfect codes in the additive channel is proved.
文摘The problem of classification of the subset of the vertices of the n-dimensional unit cube in respect to all “shifts” by a vector from Bn is studied. Some applications for the investigation of the additive channels of communication are represented.
文摘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.
文摘In the present paper, geometry of the Boolean space B<sup>n</sup> in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressions for representing of classical figures in B<sup>n</sup> and the functions of distances between them. In particular, equations in sets are considered and their interpretations in combinatory terms are given.
文摘The notion of a communication channel is one of the key notions in information theory but like the notion “information” it has not any general mathematical definition. The existing examples of the communication channels: the Gaussian ones;the binary symmetric ones;the ones with symbol drop-out and drop-in;the ones with error packets etc., characterize the distortions which take place in information conducted through the corresponding channel.
文摘In the present work, a construction making possible creation of an additive channel of cardinality s and rank r for arbitrary integers s, r, n (r≤min (n,s-1)), as well as creation of a code correcting errors of the channel A is presented.
文摘In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space B<sup>n</sup> are presented. Also, sums and multisums of various “classical figures” as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation X+Y=A, where , are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.
