Том 52, № 1 (2016)

For vector adder channel, we present an upper bound on the capacity, which coincides with the lower bound given in [1]. Thus, we prove optimality of the uniform probability distribution of symbols for γ ∈ (0, γ*] and of the twisted distribution for γ ∈ (γ*,∞), where γ is the ratio of the number of users to the number of subchannels and γ* = 1.3382.

In [1] a syndrome counting based upper bound on the minimum distance of regular binary LDPC codes is given. In this paper we extend the bound to the case of irregular and generalized LDPC codes over GF(q). A comparison with the lower bound for LDPC codes over GF(q), upper bound for the codes over GF(q), and the shortening upper bound for LDPC codes is made. The new bound is shown to lie under the Gilbert–Varshamov bound at high rates.

We study H₂/H₂/1, H₂/M/1, and M/H₂/1 queueing systems with hyperexponential arrival distributions for the purpose of finding a solution for the mean waiting time in the queue. To this end we use the spectral decomposition method for solving the Lindley integral equation. For practical application of the obtained results, we use the method of moments. Since the hyperexponential distribution law has three unknown parameters, it allows to approximate arbitrary distributions with respect to the first three moments. The choice of this distribution law is due to its simplicity and the fact that in the class of distributions with coefficients of variation greater than 1, such as log-normal, Weibull, etc., only the hyperexponential distribution makes it possible to obtain an analytical solution.

We consider flows in networks analogous to numerical flows but such that values of arc capacities are elements of a lattice. We present an analog of the max-flow min-cut theorem. However, finding the value of the maximum flow for lattice flows is based on not this theorem but computations in the algebra of matrices over the lattice; in particular, the maximum flow value is found with the help of transitive closure of flow capacity functions. We show that there exists a correspondence between flows and solutions of special-form systems of linear equations over distributive lattices.

We consider the problem of source compression under three different scenarios in the one-shot (nonasymptotic) regime. To be specific, we prove one-shot achievability and converse bounds on the coding rates for distributed source coding, source coding with coded side information available at the decoder, and source coding under maximum distortion criterion. The one-shot bounds obtained are in terms of smooth max Rényi entropy and smooth max Rényi divergence. Our results are powerful enough to yield the results that are known for these problems in the asymptotic regime both in the i.i.d. (independent and identically distributed) and non-i.i.d. settings.

We propose efficient (“fast” and low memory consuming) algorithms for universal-coding-based prediction methods for real-valued time series. Previously, for such methods it was only proved that the prediction error is asymptotically minimal, and implementation complexity issues have not been considered at all. The provided experimental results demonstrate high precision of the proposed methods.