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

Abstract

We consider the set Eⁿ of binary sequences of length n. Let M = { A₁, A₂, ... , A_s} 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(A_i, A_j) is the Hemming distance in Eⁿ.^‡ S. V. Yablonskii has posed the problem of finding an s-subset M ⊆ Eⁿ, 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) = 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 Eⁿ such that

$$\mathop {\lim }\limits_{n \to \infty } \frac{{{H_s}({M_n})}}{{{H_s}(n)}} = 1 $$

.

Original article submitted November 5, 1968.