Volume 15, Issue 5

In this paper, we construct two new families of nonlinear quinary codes, derived from the corresponding families of modified Butson–Hadamard matrices. These codes have the minimal distance close to the Plotkin bound and have very easy construction and decoding procedures.

We introduce the concepts of an information protocol on a finite set, its extension and expansion. A series of examples of manipulations of an adversary with information protocols are given. We propose a system of protection of information protocols which admits an expansion into a large number of extensions and calculate some probabilistic characteristics of its quality.

In this paper, we study the limit behaviour of the sequence of distributions of random variables taking values in the finite Abelian group (Ω, ⊕), Ω = {0, 1, . . . , k − 1}, which admit the representation. η ( N ) = f (ξ 1 , . . . ,ξ n ) ⊕ f (ξ 2 , . . . ,ξ n +1 ) ⊕ . . . ⊕ f (ξ N , . . . ,ξ N + n −1 ), where ξ 1 ,ξ 2 , . . . is the initial sequence of independent identically distributed random variables which take values in Ω, f is a k -valued function of n variables which takes values in Ω. We show that the limit behaviour of the sequence of distributions of η N as N → ∞ is determined by the minimal subgroup H of the group (Ω, ⊕) which for all x 1 , . . . , x n ∈ Ω admits the expansion f ( x 1 , . . . , x n ) ⊖ f (0, . . . ,0) ⊕ H = g ( x 1 , . . . , x n −1 ) ⊖ g ( x 2 , . . . , x n ) ⊕ H with some k -valued function g of n − 1 variables, where ⊖ is the subtraction operation in the group (Ω, ⊕). We give a description of the limit points of the sequence of distributions of the random variables η N and converging to them sequences in terms of the subgroup H and the corresponding function g .

In this paper, we study the complexity of realisation of monotone symmetric functions of algebra of logic with threshold 2 by π -circuits of closing contacts. We find the precise value of this complexity and construct the corresponding minimal circuits both in the case of unit weights of all contacts and in the case where contacts of distinct variables may be of distinct weights.

We study properties of the parameter of a Boolean function known as the affinity level, which characterises the efficiency of the linearisation method of solving systems of Boolean equations, and relations between this parameter and parameters characterising other properties of Boolean functions.

We consider the classes of functions of n variables which are intersections of the class M of monotone functions and other precomplete classes in three-valued logic. We find the asymptotic behaviour of the logarithm of cardinality of such classes as n → ∞.

We prove that the representability of a Boolean function by a repetition-free formula can be verified by a circuit of linear complexity.

We consider the class of random matrices C = ( c ij ), i, j = 1,… N , whose elements are independent random variables distributed by the same law as a certain random variable ξ such that E ξ 2 > 0. As usual, per C stands for the permanent of the matrix C. In the triangular array series where ξ = ξ N , E ξ N ≠ 0, N = 1, 2, . . . , D ξ N = o (( E ξ N ) 2 )as N → ∞, we prove that the sequence of random variables per C /( N !( E ξ N ) N ) converges in probability to one as N → ∞. A similar result is shown to be true in a more general case where the rows of the matrix C are independent N -dimensional random vectors which have the same distribution coinciding with the distribution of a random vector μ whose components are identically distributed but are, generally speaking, dependent. We give sufficient conditions for the law of large numbers to be true for the sequence per C / E per C in the cases where the vector μ coincides with the vector of frequencies of outcomes of the equiprobable polynomial scheme with N outcomes and n trials and also where μ is a random equiprobable solution of the equation k 1 + … + k N = n in non-negative integers k 1 ,…, k N .

Let S n be the symmetric group of all permutations of degree n , A be some subset of the set of natural numbers N , and T n = T n ( A ) be the set of all permutations of S n with cycle lengths belonging to A . The permutations of T n are called A -permutations. We consider a wide class of the sets A with the asymptotic density σ > 0. In this article, the limit distributions are obtained for μ m ( n )/ n as n → ∞ and m ∈ N is fixed. Here μ m ( n ) is the length of the m th maximal cycle in a random permutation uniformly distributed on T n . It is shown here that these limit distributions coincide with the limit distributions of the corresponding functionals of the random permutations in the Ewens model with parameter σ.