PROBLEMS OF INFORMATION TRANSMISSION

A translation of *Problemy Peredachi Informatsii*

Volume 44, Number 1, January–March, 2008

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** 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.