Systems Theory Research pp 25-41 | Cite as

# Asymptotically Stable Distributions of Charge on Vertices of an n-Dimensional Cube

Chapter

## Abstract

We consider the set E where p(A.

^{n}of binary sequences of length n. Let M = { A_{1}, A_{2}, ... , A_{s}} be any s-subset from E^{n}. Consider the number$$H(M) = \sum\limits_{1 \le i < j \le s} {\frac{1}{{\rho ({A_i},{A_j})}}} $$

(1)

_{i}, A_{j}) is the Hemming distance in E^{n}.^{‡}S. V. Yablonskii has posed the problem of finding an s-subset*M*⊆*E*^{ n }, in which the functional H(M) has a minimum. Physically the set M can be interpreted as a stable position of s like charged particles placed on vertices in E^{n}. In [1] this problem was completely solved for the case s(n) = 2^{n-1}. In this case it turned out that there exist two extremal sets, both of even parity. For other s, however, the question of the structure of the sets remained open. In [2] an asymptotic formulation of the problem was considered. It consists of the following. Suppose \( {H_s}(n) = \mathop {\min }\limits_{M \subseteq {E^n}} H(M) \), it is required to find a sequence { M_{n}} of s-subsets from E^{n}such that$$\mathop {\lim }\limits_{n \to \infty } \frac{{{H_s}({M_n})}}{{{H_s}(n)}} = 1 $$

## Keywords

Minimal Energy Binary Sequence Random Quantity Asymptotic Expression Dual Code
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## Literature Cited

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## Copyright information

© Consultants Bureau, New York 1973