PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii

Volume 44, Number 1, January–March, 2008
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CONTENTS

Modeling Hexagonal Constellations with Eisenstein--Jacobi Graphs
C. Martínez, E. Stafford, R. Beivide, and E. M. Gabidulin
pp. 1–11

Abstract—A set of signal points is called a hexagonal constellation if it is possible to define a metric so that each point has exactly six neighbors at distance 1 from it. As sets of signal points, quotient rings of the ring of Eisenstein–Jacobi integers are considered. For each quotient ring, the corresponding graph is defined. In turn, the distance between two points of a quotient ring is defined as the corresponding graph distance. Under certain restrictions, a quotient ring is a hexagonal constellation with respect to this metric. For the considered hexagonal constellations, some classes of perfect codes are known. Using graphs leads to a new way of constructing these codes based on solving a standard graph-theoretic problem of finding a perfect dominating set. Also, a relation between the proposed metric and the well-known Lee metric is considered.

Bent and Hyper-bent Functions over a Field of $2^\ell$ Elements
A. S. Kuz'min, V. T. Markov, A. A. Nechaev, V. A. Shishkin, and A. B. Shishkov
pp. 12–33

Abstract—We study the parameters of bent and hyper-bent (HB) functions in $n$ variables over a field $P=\mathbb{F}_q$ with $q=2^\ell$ elements, $\ell>1$. Any such function is identified with a function $F\colon Q\to P$, where $P\lt Q=\mathbb{F}_{q^n}$. The latter has a reduced trace representation $F=\operatorname{tr}^Q_P(\Phi)$, where $\Phi(x)$ is a uniquely defined polynomial of a special type. It is shown that the most accurate generalization of results on parameters of bent functions from the case $\ell=1$ to the case $\ell\gt1$ is obtained if instead of the nonlinearity degree of a function one considers its binary nonlinearity index (in the case $\ell=1$ these parameters coincide). We construct a class of HB functions that generalize binary HB functions found in [Youssef, A.M. and Gong, G., Lect. Notes Comp. Sci., vol. 2045, Berlin: Springer, 2001, pp. 406–419]; we indicate a set of parameters $q$ and $n$ for which there are no other HB functions. We introduce the notion of the period of a function and establish a relation between periods of (hyper-)bent functions and their frequency characteristics.

On the Convexity of One Coding-Theory Function
V. M. Blinovsky
pp. 34–39

Abstract—The paper completes the proof of an upper bound for multiple packings in a $q$-ary Hamming space.

Modified Sign Method for Testing the Fractality of Gaussian Noise
A. P. Kovalevskii
pp. 40–52

Abstract—Fractal Gaussian noise is a stationary Gaussian sequence of zero-mean random variables whose sums possess the stochastic self-similarity property. If the random variables are independent, the self-similarity coefficient equals $1/2$. The sign criterion for testing the hypothesis that the parameter equals $1/2$ against the alternative $H\ne 1/2$ is based on counting the sign change rate for elements of the sequence. We propose a modification of the criterion: we count sign change indicators not only for the original random variables but also for random variables formed as sums of consecutive elements. The proof of the asymptotic normality of our statistics under the alternative hypothesis is based on the theorem on the asymptotics of the covariance of sign change indicators for a zero-mean stationary Gaussian sequence with a slowly decaying correlation function.

Kalman Filter and Quantization
A. I. Ovseevich
pp. 53–71

Abstract—We give an interpretation of the problem of filtering of diffusion processes as a quantization problem. Based on this, we show that the classical Kalman–Bucy linear filter describes a flow of automorphisms of the Heisenberg algebra. We obtain new formulas for the unnormalized conditional density in the linear case, a new interpretation of the Mehler formula for the fundamental solution of the Schrödinger operator for a harmonic oscillator, and formulas for a regularized determinant of a Sturm–Liouville operator.