Minimization of Surrounding of Subsets in Hamming Space
نویسنده
چکیده
As it is well-known, the classical isoperimetric problem on the plane claims to find a simple closured curve of the plane that bounds a maximal possible area. We shell concern with some discrete analog of this problem. It was proved in [1], that for any integer m, represented in the form m = ∑r i=0 ( n i ) + δ, 0 ≤ δ < ( n r+1 ) , one of m-element subsets on n-cube B with minimum boundary is the union of the ball of radius r centered in the origin (0, ..., 0) united with the final lexicographical segment of length δ (see below). In Section 2 we are concerned with the isoperimetric type problem for the even levels of B (i.e. the collection of all vertices α of B with ‖α‖ ≡ 0 (mod 2)) and show that a ball in the even levels of B united with a final lexicographical segment (i.e. a similar construction) is a solution of it. Section 3 of this paper is devoted to isoperimetric type problems for B with some restrictions on the boundary operator. Denote by S k (α) the Hamming ball of radius k centered in the point α ∈ B. Let A ⊆ B. We call a point α ∈ A the boundary point of A iff S 1 (α) 6∈ A. The collection of all boundary points of A is called the boundary of A and denoted by Γ(A). The subset
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تاریخ انتشار 1985