Volume 31, Issue 6

A generalization of the method of C. Carlet for constructing differentially 4-uniform permutations of binary vector spaces in even dimension 2 k is suggested. It consists in restricting APN-functions in 2 k +1 variables to a linear manifold of dimension 2 k . The general construction of the method is proposed and a criterion for its applicability is established. Power permutations to which this construction is applicable are completely described and a class of suitable not one-to-one functions is presented.

The paper provides necessary and sufficient conditions for the the ergodicity of a serial connection of automata under which the output sequence of a substitution Mealy automaton is fed to the input of an output-free substitution automaton. It is shown that the condition of complete indecomposability of the state transition probability matrix of a Mealy automaton provides a sufficient condition for ergodicity of the probabilistic converter as a serial connection of automata. It is also shown that if the partial state transition functions of a Mealy automaton commute, then the condition of ergodicity of a serial connection is equivalent to that of both original probabilistic converters.

Let Γ be a distance-regular graph of diameter 3 with c 2 = 2 (any two vertices with distance 2 between them have exactly two common neighbors). Then the neighborhood Δ of the vertex w in Γ is a partial line space. In view of the Brouwer–Neumaier result either Δ is the union of isolated ( λ + 1)-cliques or the degrees of vertices k ≥ λ ( λ + 3)/2, and in the case of equality k = 5, λ = 2 and Γ is the icosahedron graph. A. A. Makhnev, M. P. Golubyatnikov and Wenbin Guo have investigated distance-regular graphs Γ of diameter 3 such that Γ 3 is the pseudo-geometrical network graph. They have found a new infinite set {2 u 2 −2 m 2 + 4 m − 3,2 u 2 −2 m 2 , u 2 − m 2 + 4 m −2;1, 2, u 2 − m 2 } of feasible intersection arrays for such graphs with c 2 = 2. Here we prove that some distance-regular graphs from this set do not exist. It is proved also that distance-regular graph with intersection array {22, 16, 5; 1, 2, 20} does not exist.

We study the complexity of implementation of the characteristic functions of spheres by contact circuits. By the characteristic functions of the sphere with center at a vertex σ̃ = ( σ 1 , …, σ n ), σ 1 , …, σ n ∈ {0, 1}, we mean the Boolean function φ σ ~ (n) $\begin{array}{} \varphi^{(n)}_{\tilde\sigma} \end{array} $( x 1 , …, x n ) which is equal to 1 on those and only those tuples of values that differ from the tuple σ̃ only in one digit. It is shown that the number 3 n − 2 of contacts is necessary and sufficient for implementation of φ σ ~ (n) $\begin{array}{} \varphi^{(n)}_{\tilde\sigma} \end{array} $( x̃ ) by a contact circuit.

For a finite q -element field F q , we established a relation between parameters characterizing the measure of affine approximation of a q -valued logic function and similar parameters for its restrictions to linear manifolds. For q > 2, an analogue of the Parseval identity with respect to these parameters is proved, which implies the meaningful upper estimates q n −1 ( q − 1) − q n /2−1 and q r −1 ( q − 1) − q r /2−1 , for the nonlinearity of an n -place q -valued logic function and of its restrictions to manifolds of dimension r . Estimates characterizing the distribution of nonlinearity on manifolds of fixed dimension are obtained.

We consider polynomials f ( x 1 , …, x n ) over a finite field that possess the following property: for some element b of the field the set of solutions of the equation f ( x 1 , …, x n ) = b coincides with the set of solutions of some system of linear equations over this field. Such polynomials are said to be multiaffine with respect to the right-hand side b . We obtain the properties of multiaffine polynomials over a finite field. Then we show that checking the multiaffinity with respect to a given right-hand side may be done by an algorithm with polynomial (in terms of the number of variables and summands of the input polynomial) complexity. Besides that, we prove that in case of the positive decision a corresponding system of linear equations may be recovered with complexity which is also polynomial in terms of the same parameters.

We consider the probabilities of large deviations for the branching process Z n in a random environment, which is formed by independent identically distributed variables. It is assumed that the associated random walk S n = ξ 1 + … + ξ n has a finite mean μ and satisfies the Cramér condition E e hξ i < ∞, 0 < h < h + . Under additional moment constraints on Z 1 , the exact asymptotic of the probabilities P (ln Z n ∈ [ x , x + Δ n )) is found for the values x/n varying in the range depending on the type of process, and for all sequences Δ n that tend to zero sufficiently slowly as n → ∞. A similar theorem is proved for a random process in a random environment with immigration.