Volume 26, Issue 5

Let GF ( q ) be the field of q elements and V n ( q ) denote the n -dimensional vector space over the field GF ( q ). The linearized polynomial that corresponds to the polynomial f(x)=xn−∑i=0n−1cixi$f(x) = {x^n} - \sum\limits_{i = 0}^{n - 1} {{c_i}{x^i}} \;$ over the field GF ( q ) is the polynomial F(x)=xqn−∑i=0n−1cixqi$F(x) = {x^{{q^n}}} - \sum\limits_{i = 0}^{n - 1} {{c_i}{x^{{q^i}}}}$. Let T f denote the transformation of the vector space V n ( q ) determined by the rule Tf(u0,...,un−2,un−1)=(u1,...,un−1,∑i=0n−1ciui)${T_f}\left( {({u_0},...,{u_{n - 2}},{u_{n - 1}})} \right) = ({u_1},...,{u_{n - 1}},\sum\limits_{i = 0}^{n - 1} {{c_i}{u_i}} )$. It is shown that if c 0 ≠ 0, then the graph of the transformation T f is isomorphic to the graph of the transformation Q : α → α q on the set of all roots of the polynomial F ( x ) in its splitting field. In this case the graph of the transformation T f consists of cycles of lengths 1 ≤ d 1 ≤ d 2 ≤ ... ≤ d r if and only if the polynomial F ( x ) is the product of r + 1 irreducible factors of degrees 1, d 1 , d 2 , ... d r .

We obtain achievable lower and upper bounds for the sums of modules of Walsh coefficients of Boolean functions of nvariables. An average value of such sums in the class of all Boolean functions of n variables and in its subclass consisting of all balanced functions is evaluated. We present some classes of nonlinear balanced functions whose sums of modules of Walsh coefficients are close to the obtained lower and upper bounds

We propose and investigate new algorithms permitting to find an identical object in the database using the number of operations not depending on the volume of the database. One algorithm requires memory size that depends linearly on the database volume in the average.

The paper is concerned with the problem of synthesis of two-pole contact circuits implementing n -place Boolean functions and admitting short fault detection and diagnostic tests with respect to closures of contacts. It is shown that almost all n -place Boolean functions are implemented by irredundant two-pole contact circuits admitting single fault detection, complete fault detection and single diagnostic tests of constant length. We also prove that: 1) any Boolean function f ( x 1 ,..., x n ) may be implemented by an irredundant two-pole contact circuit containing at most one input variable distinct from the variables x 1 ,..., x n and admitting single and complete fault detection tests of length at most 2 n ; 2) any Boolean function f ( x 1 ,..., x n ) may be implemented by an irredundant two-pole contact circuit containing at most two input variables distinct from the variables x 1 ,..., x n and admitting single diagnostic test of length at most 4 n .

Properties of the orbital derivatives over subgroups of the group G n generated by the additive groups of the residue ring 𝕫 2 n and the n -dimensional vector space V n over the field GF (2) are considered. Nonrefinable sequences of nested orbits for the subgroups of the group G n and of the Sylow subgroup P n of the symmetric group S 2 n are described. For the orbital derivatives, three analogs of the concept of the degree of nonlinearity for functions over 𝕫 2 n or V n are suggested.