Following Schumacher and Westmoreland, we address the problem of the capacity of a quantum wiretap channel. We first argue that, in the definition of the so-called “quantum privacy,” Holevo quantities should be used instead of classical mutual informations. The argument actually shows that the security condition in the definition of a code should limit the wiretapper’s Holevo quantity. Then we show that this modified quantum privacy is the optimum achievable rate of secure transmission.

Characteristics of a successive cancellation scheme, widely used in iterative decoding, is investigated. Comparison with another popular method, the minimum mean-square error (MMSE) method, is also provided.

A Steiner quadruple system SQS(v) of order v is a 3-design T (v; 4; 3;λ) with λ = 1. In this paper we describe all nonisomorphic systems SQS(16) that can be obtained by the generalized concatenated construction (GC-construction). These Steiner systems have rank at most 13 over F2. In particular, there is one system SQS(16) of rank 11 (points and planes of the a fine geometry AG(4; 2)), fifteen systems of rank 12, and 4131 systems of rank 13. All these Steiner systems are resolvable.

A new statistical test is proposed for testing the hypothesis H 0 that symbols of an alphabet are generated with equal probabilities against the alternative hypothesis H 1, the negation of H 0. The new method is applied to testing generators of pseudorandom numbers. It is experimentally demonstrated that the method makes it possible to detect deviations from randomness for many generators which “withstand” previously known statistical tests.

A (w, r)-cover-free code is the incidence matrix of a family of sets where no intersection of w members of the family is covered by the union of r others. We obtain a new condition in view of which (w, r)-cover-free codes with a simple structure are optimal. We also introduce (w, r)-cover-free codes with a constraint set.

We establish the properness of some classes of binary block codes with symmetric distance distribution, including Kerdock codes and codes that satisfy the Grey-Rankin bound, as well as the properness of Preparata codes, thus augmenting the list of very few known proper nonlinear codes.

Characteristics of a successive cancellation scheme, widely used in iterative decoding, is investigated. Comparison with another popular method, the minimum mean-square error (MMSE) method, is also provided.

A Steiner quadruple system SQS(v) of order v is a 3-design T (v, 4, 3, ) with = 1. In this paper we describe all nonisomorphic systems SQS(16) that can be obtained by the generalized concatenated construction (GC-construction). These Steiner systems have rank at most 13 over $$\mathbb{F}$$ 2. In particular, there is one system SQS(16) of rank 11 (points and planes of the a fine geometry AG(4, 2)), fifteen systems of rank 12, and 4131 systems of rank 13. All these Steiner systems are resolvable.

Under the assumption that the distribution function of the observation noise is known, both for the case of a predefined observation design and the case where observation designing is possible, we construct estimates of smooth functionals of the regression function, for which lower bounds on mean-square risks of arbitrary estimates of smooth functionals obtained in [1, 2] are asymptotically attained.

Following Schumacher and Westmoreland, we address the problem of the capacity of a quantum wiretap channel. We first argue that, in the definition of the so-called quantum privacy, Holevo quantities should be used instead of classical mutual informations. The argument actually shows that the security condition in the definition of a code should limit the wiretappers Holevo quantity. Then we show that this modified quantum privacy is the optimum achievable rate of secure transmission.

We establish the properness of some classes of binary block codes with symmetric distance distribution, including Kerdock codes and codes that satisfy the Grey-Rankin bound, as well as the properness of Preparata codes, thus augmenting the list of very few known proper nonlinear codes.

In this paper, we address the problem of image denoising using a stochastic differential equation approach. Proposed stochastic dynamics schemes are based on the property of diffusion dynamics to converge to a distribution on global minima of the energy function of the model, under a special cooling schedule (the annealing procedure). To derive algorithms for computer simulations, we consider discrete-time approximations of the stochastic differential equation. We study convergence of the corresponding Markov chains to the diffusion process. We give conditions for the ergodicity of the Euler approximation scheme. In the conclusion, we compare results of computer simulations using the diffusion dynamics algorithms and the standard Metropolis Hasting algorithm. Results are shown on synthetic and real data.

A convolutional code can be used to detect or correct infinite sequences of errors or to correct infinite sequences of erasures. First, erasure correction is shown to be related to error detection, as well as error detection to error correction. Next, the active burst distance is exploited, and various bounds on erasure correction, error detection, and error correction are obtained for convolutional codes. These bounds are illustrated by examples.

A mathematical model of an adaptive random multiple access communication network is investigated. The value of the network critical load is found; in the critical load, asymptotic probability distributions for states of the information transmission channel and for the number of requests in the source of repeated calls are found. It is proved that distributions of the normalized number of requests belong to the class of normal and exponential distributions, and it is shown how conditional normal distributions pass in the limit to the class of exponential ones.

We consider the 2 statistic, destined for testing the symmetry hypothesis, which has the form n2 = n -[Fn(x) + Fn(-x) - 1]2 dFn(x), where Fn(x) is the empirical distribution function. Based on the Laplace method for empirical measures, exact asymptotic (as n ) of the probability Pr{n2 > nv} for 0 < v < 1/3 is found. Constants entering the formula for the exact asymptotic are computed by solving the extreme value problem for the rate function and analyzing the spectrum of the second-order differential equation of the Sturm Liouville type.

An infinite-server queueing system is considered where access of customers to service is controlled by a gate. The gate is open only if all servers are free. Otherwise, customers are put on a queue. Asymptotic behavior of the system in heavy traffic is studied under the assumption that the service time distribution has a power tail.

We consider the ω2 statistic, destined for testing the symmetry hypothesis, which has the form $$\omega _n^{\text{2}} = n\;\int\limits_{ - \infty }^\infty {[F_n (x)\; + F_n ( - x)\; - 1]^2 dF_n (x),}$$ where F n (x) is the empirical distribution function. Based on the Laplace method for empirical measures, exact asymptotic (as n → ∞) of the probability $$P\{ \omega _n^2 > nv\} $$ for 0 < v < 1/3 is found. Constants entering the formula for the exact asymptotic are computed by solving the extreme value problem for the rate function and analyzing the spectrum of the second-order differential equation of the Sturm–Liouville type.

We consider the optical-acoustic tomography problem. In the general case, the problem is to reconstruct a real-valued function with a compact support in the n-dimensional Euclidean space via its spherical integrals, i.e., integrals over all (n - 1)-dimensional spheres centered at points on some (n - 1)-dimensional hypersurface. We deal with the cases n = 2 and n = 3, which are of the most practical interest from the standpoint of possible medical applications. We suggest a new effective method of reconstruction, develop restoration algorithms, and investigate the quality of the algorithms for these cases. The main result of the paper is construction of explicit approximate reconstruction formulas; from the mathematical standpoint, these formulas give the parametrix for the optical-acoustic tomography problem. The formulas constructed is a background for the restoration algorithms. We performed mathematical experiments to investigate the quality of the restoration algorithms using the generally accepted tomography quality criteria. The results obtain lead to the general conclusion: the quality of the restoration algorithms developed for optical-acoustic tomography is only slightly lower then the quality of the convolution and back projection algorithm used in Radon tomography, which is a standard de facto.

The paper considers the problem of estimating a signal with finitely many points of discontinuity from observations against white Gaussian noise. It is shown that, with an appropriate choice of a generator polynomial, an estimation method based on wavelets yields asymptotically minimax (up to a constant) estimates for functions sufficiently smooth outside the discontinuity points.