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

  • V. K. Leont’ev


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
$$H(M) = \sum\limits_{1 \le i < j \le s} {\frac{1}{{\rho ({A_i},{A_j})}}} $$
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
$$\mathop {\lim }\limits_{n \to \infty } \frac{{{H_s}({M_n})}}{{{H_s}(n)}} = 1 $$


Minimal Energy Binary Sequence Random Quantity Asymptotic Expression Dual Code 
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Copyright information

© Consultants Bureau, New York 1973

Authors and Affiliations

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

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