Cybernetics and Control Theory

B. S. Zilberman

On the Arrangement of Charges at the Vertices of the Unit \(n\)-Dimensional Cube*

(Presented by Academician P. S. Novikov on 11 X 1962)

We consider the set \(E_n\) of all vertices of the unit \(n\)-dimensional cube, i.e., the set of binary strings \((\sigma_1,\ldots,\sigma_n)\) of length \(n\). The set \(E_n\) becomes a metric space if, for any pair of its elements \(A=(\alpha_1,\ldots,\alpha_n)\) and \(B=(\beta_1,\ldots,\beta_n)\), the distance is defined by
\(\rho(A,B)=|\alpha_1-\beta_1|+|\alpha_2-\beta_2|+\cdots+|\alpha_n-\beta_n|\) \((^1)\).

To each set of vertices \(M=\{A_1,\ldots,A_m\}\) \((m\geq 2)\) there is assigned a number \(H(M)\), called the energy of the given set:

\[ H(M)=\sum_{1\leq i<j\leq m}\frac{1}{\rho(A_i,A_j)}. \]

The problem is, among all sets of \(m\) vertices of the unit \(n\)-dimensional cube \((2\leq m\leq 2^n)\), to find a set with minimal energy.

If the number \(H(M)\) is taken as the value of the energy of a system of equally charged particles located at the vertices \(A_1,\ldots,A_m\), then this problem can be given a physical meaning: to place \(m\) equally charged particles at the vertices of the unit \(n\)-dimensional cube so that the resulting system of charges is stable**.

The problem of arranging charges at the vertices of the unit \(n\)-dimensional cube arose in connection with the problem of constructing optimal self-correcting codes \((^1)\). For this it is necessary to consider the coding problem in a somewhat different formulation. Namely: to find such a system of \(m\) code points in the unit \(n\)-dimensional cube that the probability of detecting and correcting an error (for a given noise source) is greatest. There is an opinion that every solution of the charge problem will be a solution of the coding problem. In that case it will be possible to construct optimal codes using physical principles.

A complete solution of the charge problem has proved difficult. The author investigated the case when \(m=2^{n-1}\). In the present note*** it is shown that in this case there exist only two solutions, namely, the sets whose characteristic functions are the linear functions of \(n\) variables
\(x_1+\cdots+x_n \pmod 2\) and \(x_1+\cdots+x_n+1 \pmod 2\). Consequently, for this special case the supposition stated above has been fully confirmed.

Let

\[ g(n)=C_n^1/1+C_n^2/2+\cdots+C_n^n/n,\qquad \overline{M}=E_n\setminus M. \]

It is easily proved

Theorem 1.

\[ H(\overline{M})-H(M)=\left(2^{n-1}-m\right)g(n). \tag{1} \]

Theorem 1 makes it possible to restrict ourselves to consideration of the case \(m\leq 2^{n-1}\). Number all points of the cube \(E_n=\{A_1,\ldots,A_{2^n}\}\) arbitrarily. Let us partition the cube into two sets \(M\) and \(\overline{M}=E_n\setminus M\). Denote by

* The work was reported at the seminar on discrete analysis at Moscow State University named after M. V. Lomonosov in the spring of 1961.

** This problem was posed and suggested to the author by S. V. Yablonskii as a thesis topic.

*** The present note is a development of the 1960 thesis.

\(a_i^{(s)}\) be the number of points lying at distance \(s\) from the point \(A_i\) and belonging to the same set (\(M\) or \(\overline M\)) as \(A_i\). With the aid of these numbers one obtains a simple expression for the energy

\[ H(M)=\frac14\sum_{s=1}^n \frac1s \sum_{i=1}^{2^n} a_i^{(s)} -\frac12(2^{n-1}-m)g(n). \tag{2} \]

Lemma 1. The numbers \(a_i^{(s)}\) satisfy the relations:

\[ \sum_{i=1}^{2^n} a_i^{(1)}a_i^{(s)} +\sum_{i=1}^{2^n} \bigl(n-a_i^{(1)}\bigr)\bigl(C_n^s-a_i^{(s)}\bigr) \]

\[ =(n-s+1)\sum_{i=1}^{2^n} a_i^{(s-1)} +(s+1)\sum_{i=1}^{2^n} a_i^{(s+1)} \tag{3} \]

\[ (s=1,\ldots,n-1),\qquad a_i^{(0)}=1. \]

The numbers \(a_i^{(s)}\), as is easy to see, also satisfy the conditions

\[ \sum_{i=1}^{2^n}\sum_{s=1}^n a_i^{(s)} = m(m-1)+(2^n-m-1)(2^n-m), \]

where \(m\) is the cardinality of \(M\). In what follows we everywhere assume that \(m=2^{n-1}\).

Consider the system of equations in the \(a_i^{(s)}\) \((i=1,\ldots,2^n;\ s=1,\ldots,n)\)

\[ \sum_{i=1}^{2^n} a_i^{(1)}a_i^{(s)} +\sum_{i=1}^{2^n} \bigl(n-a_i^{(1)}\bigr)\bigl(C_n^s-a_i^{(s)}\bigr) \]
\[ -(n-s+1)\sum_{i=1}^{2^n} a_i^{(s-1)} -(s+1)\sum_{i=1}^{2^n} a_i^{(s+1)} =0 \quad (s=1,\ldots,n-1); \tag{4} \]

\[ \sum_{i=1}^{2^n}\sum_{s=1}^n a_i^{(s)} -2^n(2^{n-1}-1)=0. \]

Obviously, every partition of the cube \(E_n\) into two sets of equal cardinality gives rise to a solution of system (4).

We shall call a solution of system (4) homogeneous if it has
\(a_i^{(1)}=a_j^{(1)}=a^{(1)}\) \((i,j=1,\ldots,2^n)\). A set \(M\) of vertices of the cube \(E_n\) will be called homogeneous if the corresponding partition of the cube gives rise to a homogeneous solution. Let us consider an important class of homogeneous sets.

A parity counter of order \(k\), \(E_n^k\) \((k=0,1,\ldots,n-1)\), is a set whose characteristic function is
\(f(x_1,\ldots,x_n)=x_{j_1}+\cdots+x_{j_{n-k}}+c\pmod 2\), \(c\in\{0,1\}\).

It is easy to see that for the parity counter \(E_n^k\)

\[ a_i^{(s)}=a_j^{(s)}=p_k^{(s)} \quad (i,j=1,\ldots,2^n;\ s=1,\ldots,n), \]

where

\[ p_k^{(s)}=C_k^s+C_k^{s-2}C_{n-k}^2+C_k^{s-4}C_{n-k}^4+\cdots \tag{5} \]

In particular, \(a_i^{(1)}=a_j^{(1)}=k\); consequently, the set \(E^k\) is homogeneous.

Denote by \(a^{(s)}\) the quantity \(2^{-n}\sum_{i=1}^{2^n} a_i^{(s)}\).

Lemma 2. For a homogeneous solution one has \(a^{(s)}=p_k^{(s)}\) \((s=1,\ldots,n)\).

Proof. For a homogeneous solution, system (4) has the form

\[ (s+1)a^{(s+1)}=2a^{(1)}a^{(s)}-na^{(s)}-(n-s+1)a^{(s-1)}- \]
\[ {}-C_n^s a^{(1)}+nC_n^s \qquad (s=1,\ldots,n-1), \]

\[ a^{(1)}+a^{(2)}+\cdots+a^{(n)}=2^{n-1}-1. \tag{6} \]

It is clear from this that \(a^{(s+1)}\) is expressed as a polynomial in \(a^{(1)}\) of degree \(s+1\). Substituting, in the last of equations (6), their expressions through \(a^{(1)}\) in place of \(a^{(2)},\ldots,a^{(n)}\), we obtain a polynomial of degree \(n\) in \(a^{(1)}\). Consequently, system (6) has \(n\) solutions. But the \(n\) distinct solutions of system (6) generate the sets \(E_n^k\) \((k=0,1,\ldots,n-1)\), and therefore \(a^{(s)}=p_k^{(s)}\) \((s=1,\ldots,n)\). The lemma is proved.

Corollary. The energy of a homogeneous solution, defined as

\[ \frac14\sum_{s=1}^n \frac1s \sum_{i=1}^{2^n} a_i^{(s)}, \]

is equal to the energy of one of the parity counters.

Lemma 3. Among all parity counters, the set \(E_n^0\) has the minimal energy.

Proof. By virtue of (5), the energy of the set \(E_n^k\) is equal to

\[ H(E_n^k)=2^{n-2}\sum_{s=1}^n \frac1s\,p_k^{(s)}. \]

Let

\[ P_k(x)=\frac{(1+x)^{n-k}+(1-x)^{n-k}}{2}(1+x)^k-1 \quad\text{and}\quad Q_k(x)=\int_0^x \frac{P_k(x)}{x}\,dx. \]

Obviously,

\[ P_k(x)=\sum_{s=1}^n p_k^{(s)}x^s \quad\text{and}\quad Q_k(x)=\sum_{s=1}^n \frac{p_k^{(s)}}{s}x^s. \]

Further,

\[ H(E_n^k)=2^{n-2}Q_k(1) =2^{n-2}\int_0^1 \frac{(1+x)^n+(1+x)^n\left(\frac{1-x}{1+x}\right)^{n-k}-2}{2x}\,dx. \]

Hence it follows immediately that

\[ H(E_n^0)<H(E_n^1)<\cdots<H(E_n^{\,n-1}). \tag{7} \]

The lemma is proved.

Lemma 4. For any solution of system (4): 1) \(a^{(2)}\geqslant 0\); 2) \(0\leqslant a^{(1)}\leqslant n\), if \(a^{(2)}\leqslant C_n^2\).

Proof. Equations (4) for \(s=1\) give

\[ 2\sum_{i=1}^{2^n} a_i^{(2)} = 2\sum_{i=1}^{2^n} a_i^{(1)2} -2n\sum_{i=1}^{2^n} a_i^{(1)} +n^2\cdot 2^n-n\cdot 2^n. \]

We shall have

\[ a^{(2)}\geqslant a^{(1)2}-na^{(1)}+C_n^2 \quad (\text{for } n\geqslant 2), \]

and also \(0\leqslant a^{(1)}\leqslant n\), if \(a^{(2)}\leqslant C_n^2\). The lemma is proved.

Denote by \(F_s\) \((s=1,2,\ldots,n-1)\), \(F_0\) the left-hand sides of equations (4), and by \(G_r\) the quantity

\[ \sum_{i=1}^{2^n} a_i^{(r)} \quad (r=1,\ldots,n). \]

We shall seek the minimum of the function \(H' = 4H = \sum_{s=1}^{n} \dfrac{1}{s}\sum_{i=1}^{2^n} a_i^{(s)}\) on the manifold defined by the equations \(F_0 = 0, F_1 = 0, \ldots, F_{n-1} = 0\) and the boundary conditions \(G_1 \geqslant 0, G_3 \geqslant 0, G_4 \geqslant 0, \ldots, G_n \geqslant 0\). In view of these boundary conditions and Lemma 4, the function \(H'\) is bounded from below and hence, being continuous and defined on a closed manifold, attains its minimum value.

Lemma 5. For \(n > 3\), the minimum of the function \(H'\) is attained at a homogeneous solution.

Proof. We apply the method of Lagrange multipliers. Consider the function
\[ \Phi = H' + \lambda_0 F_0 + \sum_{s=1}^{n-1} \lambda_s F_s + \sum_{r=1,\ r\ne 2}^{n} \mu_r \sigma_r G_r, \]
where \(\lambda_0, \lambda_1, \ldots, \lambda_{n-1}, \mu_1, \mu_3, \mu_4, \ldots, \mu_n\) are Lagrange multipliers, \(\sigma_r \in \{0,1\}\). (\(\sigma_r\) is introduced so as not to consider separately the different cases of the position of a stationary point: to each set \((\sigma_1,\sigma_3,\sigma_4,\ldots,\sigma_n)\), where not all \(\sigma_r\) are equal to 0, there corresponds a certain position on the boundary; the set \((0,0,0,\ldots,0)\) corresponds to an interior point.) We exclude from consideration the set \((1,1,1,\ldots,1)\), since in this case the energy is obviously not minimal for \(n>3\). Indeed, then
\[ \sum_{i=1}^{2^n} a_i^{(2)} = 2^n(2^{n-1}-1) \]
and
\[ H' = \frac12 \sum_{i=1}^{2^n} a_i^{(2)} > 4H(E_n^0) \quad (n>3). \]

Let us write the Lagrange equations for the function \(\Phi\):
\[ \begin{aligned} \partial\Phi/\partial a_i^{(1)} &= 1+\lambda_0+\sigma_1\mu_1 +2\lambda_1(2a_i^{(1)}-n) +\lambda_2(2a_i^{(2)}-C_n^2-n+1) \\ &\quad+\lambda_3(2a_i^{(3)}-C_n^3)+\cdots +\lambda_{n-1}(2a_i^{(n-1)}-C_n^{n-1})=0,\\ \partial\Phi/\partial a_i^{(2)} &= 1/2+\lambda_0-2\lambda_1 +\lambda_2(2a_i^{(1)}-n)-\lambda_3(n-2)=0,\\ \partial\Phi/\partial a_i^{(3)} &= 1/3+\lambda_0+\sigma_3\mu_3 -3\lambda_2+\lambda_3(2a_i^{(1)}-n)-\lambda_4(n-3)=0,\\ &\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ \partial\Phi/\partial a_i^{(l)} &= 1/l+\lambda_0+\sigma_l\mu_l -l\lambda_{l-1}+\lambda_l(2a_i^{(1)}-n)-\lambda_{l+1}(n-l)=0,\\ &\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \tag{8}\\ \partial\Phi/\partial a_i^{(n-1)} &= 1/(n-1)+\lambda_0+\sigma_{n-1}\mu_{n-1} -(n-1)\lambda_{n-2}+\lambda_{n-1}(2a_i^{(1)}-n)=0,\\ \partial\Phi/\partial a_i^{(n)} &= 1/n+\lambda_0+\sigma_n\mu_n-n\lambda_{n-1}=0\\ &\hspace{2.5cm}(i=1,\ldots,2^n). \end{aligned} \]

From equations (8) and the condition \(\sigma_l=0\) (\(l\ne 2\)) it follows that \(a_i^{(1)}=a_j^{(1)}\) \((i,j=1,\ldots,2^n)\). The lemma is proved.

Comparing Lemmas 2, 3, and 5, and noting that the proposition stated below is obvious for \(n\leqslant 3\), we obtain Theorem 2.

Theorem 2. The minimum energy of a set of cardinality \(2^{n-1}\) is attained on the parity counter \(E_n^0\) and is equal to
\[ H(E_n^0)=2^{n-2}\left(C_n^2/2+C_n^4/4+\cdots\right). \]

The uniqueness of the set with minimal energy follows from the uniqueness of the set for which \(a_i^{(1)}=0\) \((i=1,\ldots,2^n)\).

In an analogous way, using inequalities (7) and the second assertion of Lemma 4, one could show that the maximum of the energy is attained on the parity counter \(E_n^{\,n-1}\), i.e., on the set whose characteristic function is equal to \(x_j\) \((j=1,\ldots,n)\).

Moscow Electromechanical Institute

Received
14 IX 1962

REFERENCES

1. R. W. Hamming, in: Codes with Error Detection and Correction, IL, 1956, p. 7.