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

• V. K. Leont’ev

## Abstract

We consider the set En of binary sequences of length n. Let M = { A1, A2, ... , As} be any s-subset from En. Consider the number
$$H(M) = \sum\limits_{1 \le i < j \le s} {\frac{1}{{\rho ({A_i},{A_j})}}}$$
(1)
where p(Ai, Aj) is the Hemming distance in En. S. V. Yablonskii has posed the problem of finding an s-subset ME 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 En. In [1] this problem was completely solved for the case s(n) = 2n-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 { Mn} of s-subsets from En 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Literature Cited

1. 1.
B. S. ZiPberman, “On the distribution of charge in the vertices of the unit n-dimensional cube,” Dokl. Akad. Nauk, Vol. 149, No. 3 (1963).Google Scholar
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T.N. Kruglova, “On asymptotic methods of solving problem on charges,” in: Problemy Kibernetiki, Vol. 13, Moscow (1965).Google Scholar
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W. Peterson, Error Correcting Codes, Wiley, New York (1961).
4. 4.
G. Polya and G. Szego, Aufgaben und Lehrsätze aus der Analysis, Springer, Leipzig (1935).Google Scholar
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D. D. Joshi, “A note on upper bounds for minimum distance codes,” Inf. Control, 1(3):289–295 (1958).
6. 6.
J. MacWilliams, “The structure and properties of binary cyclic alphabets,” Bell. Syst. Techn. Journ., 44(2):303–332 (1965).

© Consultants Bureau, New York 1973

## Authors and Affiliations

• V. K. Leont’ev
• 1
1. 1.NovosibirskRussia