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Let p ≥ 3 be a prime number. In this paper we construct two families of nonlinear p -ary codes derived from the corresponding families of modified Butson–Hadamard matrices. These codes have the minimal distances close to the Plotkin bound and very easy construction and decoding procedures....
We consider the random variable equal to the number of trees of given size in the graph of a random one-to-one mapping of an n -element set into itself with m connected components. We obtain limit theorems which describe the distribution of this characteristic in the case where n → ∞, m / ln...
The algebraic method is widely used in analysis of filter generators of pseudo-random sequences. It is based on obtaining low-degree Boolean equations in bits of the initial states of the generator. The problem to obtain such equations reduces to finding low-degree annihilators for the filtering...
We define the distance d ( z , z ′) between points z = x + iy and z ′ = x ′ + iy ′ in the upper half-plane, setting where u = | z – z ′| 2 /( yy ′). The circle K ( z 0 , T ) with centre in a point z 0 consists of the points z satisfying the inequality d ( z , z 0 ) ≤ T . Let N ( z...
Boolean functions have found a widespread use in cryptography. In connection with the advent of the 'algebraic' attack on stream ciphers, the Boolean functions, which are used in these ciphers as nonlinear filters, have to possess, among other properties, high algebraic immunity. One more...
We introduce a class of cell circuits, T -circuits, and describe a connection between the lower bound for the area and the depth of the circuits of this class: the less the depth the greater the area of a circuit. We give examples of T -circuits with logarithmic depth in the problem of...
We suggest a new approach to the study of the structural properties of random permutations based on defining some general parametric measure on the set S n of all permutations of degree n which assigns to a permutation with cyclic structure a = ( a 1 , . . . , a n ) the probability proportional...
We give an example of a quasi-universal function in the class ϵ 2 of the Grzegorczyk hierarchy. This function is of very simple structure and does not contain an explicit enumeration of any Turing machine. As a corollary we obtain a simple basis over superposition in the class ϵ 2 .
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