TITLE:
On Addition of Sets in Boolean Space
AUTHORS:
Vladimir Leontiev, Garib Movsisyan, Zhirayr Margaryan
KEYWORDS:
Hadamard Matrices, Minkowski Addition, Multiset, Cardinality, Multisum, Interval, Quadrate, Boolean Space, Stabilizer, Additive Channel
JOURNAL NAME:
Journal of Information Security,
Vol.7 No.4,
July
19,
2016
ABSTRACT: 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 Bn 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.