ELECTRICAL ENGINEERING

Yu. L. SAGALOVICH

ON THE NUMBER OF SYMMETRY TYPES OF CONTACT \((1,k)\)-POLES

(Presented by Academician V. S. Kulebakin, 10 IV 1959)

1. There are in all \(Q=C_\mu^k\) functionally distinct oriented contact \((1,k)\)-poles realizing a basic sequence \({}^{1}\) of Boolean functions \(f_1, f_2,\ldots,f_k\) (the numbering is immaterial) of \(n\) variables \(x_1,x_2,\ldots,x_n\), where \(\mu=2^{2^n}-2\). All \((1,k)\)-poles that can be obtained from a given one by transformations from the group \(O_n\) of order \(2^n n!\) of the hypercube belong to one type. Let \(s\in O_n\) induce a permutation \(S\) of degree \(2^n\) of the unit constituents \(\sigma_i\) \((i=0,1,\ldots,2^n-1)\), a permutation \(t\) of degree \(\mu\) of the Boolean functions \(f\not\equiv \mathrm{const}\) of \(n\) variables, and a permutation \(T\) of degree \(Q\) of the \((1,k)\)-poles: \(s\to S\to t\to T\).

The set of \(2^n n!\) permutation \(Q\times Q\)-matrices \(A_T\) will be a representation \(\Gamma\) of the group \(O_n\), which is reducible and contains the identity representation as many times as there are types of \((1,k)\)-poles. Hence the number of types of \((1,k)\)-poles is

\[ N_{n,k}=\frac{1}{2^n n!}\sum_c n_c\chi(c), \]

where \(n_c\) is the number of elements of the class \(C\); \(\chi(c)\) is the character of the class \(C\) of the group \(O_n\) in the representation \(\Gamma\).

If \((t_1,t_2,\ldots,t_\nu)\) is the cycle structure of the permutation \(t\) \((\nu<2^{e^n/e})\), i.e.

\[ \sum_{i=1}^{\nu} i t_i=\mu;\qquad \sum_{i=1}^{m} i b_i=k \quad (m=1,2,\ldots,k),\qquad s\in C, \]

then

\[ \chi(c)=\sum \prod C_{t_i}^{b_i}; \]

the sum is taken over all decompositions \((b_1,b_2,\ldots,b_m)\) of the number \(k\) into positive summands, and \(C_{t_i}^{b_i}\) is computed only for those \(i\) that satisfy the condition \(b_i\le t_i\). To determine the cycle structure \((t_1,t_2,\ldots,t_\nu)\) we use the fact that the unit constituents stand in the same relation to all functions of logic algebra as the variables do to the constituents. It follows from this that if some cycle \((x_{i_1}x_{i_2}\ldots x_{i_p})\) induces a permutation of \(2^p\) constituents of the form \(x_{i_1}^{\varepsilon_1}x_{i_2}^{\varepsilon_2}\ldots x_{i_p}^{\varepsilon_p}\), with cycle structure \((S_1,S_2,\ldots,S_p)\), then the cycle \((\sigma_{i_1}\sigma_{i_2}\ldots\sigma_{i_p})\) induces a permutation of all \(2^p\) Boolean functions of the form \(\bigvee_{j=1}^{p}\sigma_{i_j}^{\varepsilon_j}\) (where, as usual, \(\varepsilon_j=1\) or \(0\)), whose cycle structure is \((\lambda_1,\lambda_2,\ldots,\lambda_p)\), and \(\lambda_1=S_1,\ \lambda_2=S_2,\ldots,\lambda_p=S_p\). To the cycle structure \((\lambda_1,\lambda_2,\ldots,\lambda_p)\)

we carry the expression

\[ P(p)=\sum_{j=1}^{p}\lambda_j \tau_j . \]

Then (as in (2)) the cyclic structure of the substitution \(t\) will be characterized by the expression

\[ P=\prod_q P(p_q)=\prod_q\left(\sum_{j=1}^{p_q}\lambda_j \tau_j\right) = t'_1\tau_1+t_2\tau_2+\cdots+t_\nu\tau_\nu, \]

where

\[ t'_1=t_1+2,\qquad \sum_q p_q=2^n,\qquad \tau_{j_1}\tau_{j_2}=j_1j_2/j_3\cdot \tau_{j_3}, \]

and \(j_3\) is the least common multiple of the numbers \(j_1\) and \(j_2\).

The computations gave the following results:

For \(k=1\), result (2) is obtained by increasing all numbers of the corresponding graph by 2.

1. Any \((1,k)\)-pole with \(n\) variables converts \(n\)-valued code combinations into \(k\)-valued ones. Therefore, for \(k<n\), \(N_{n,k}\) is also the number of types of transformations resembling the processes for decoding combinations of a redundant code.

2. It is easy to show that

\[ N_{n,k}\sim \frac{Q}{2^n n!}. \]

Laboratory of Information Transmission Systems
Academy of Sciences of the USSR

Received
10 IV 1959

REFERENCES

1. G. N. Povarov, DAN, 100, No. 5, 909 (1955).
2. D. Slepian, Canad. J. Math., 5, No. 2, 185 (1953).