PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii

Volume 42, Number 1, January–March, 2006
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CONTENTS

A New Class of Quaternary Codes
S. A. Stepanov
pp. 1–9

Abstract—In this paper we introduce a new concept of modified Butson–Hadamard matrices and construct two families of quaternary codes derived from the corresponding families of modified complex matrices with entries from a finite cyclic group of order 4. These nonlinear codes have parameters lying very close to the Plotkin bound and admit very easy construction and decoding procedures.

Vasil'ev Codes of Length $n=2^m$ and Doubling of Steiner Systems $S(n,4,3)$ of a Given Rank
V. A. Zinoviev and D. V. Zinoviev
pp. 10–29

Abstract—Extended binary perfect nonlinear Vasil'ev codes of length $n=2^m$ and Steiner systems $S(n,4,3)$ of rank $n-m$ over $\mathbb{F}_2$ are studied. The generalized concatenated construction of Vasil'ev codes induces a variant of the doubling construction for Steiner systems $S(n,4,3)$ of an arbitrary rank $r$ over $\mathbb{F}_2$. We prove that any Steiner system $S(n=2^m,4,3)$ of rank $n-m$ can be obtained by this doubling construction and is formed by codewords of weight $4$ of these Vasil'ev codes. The length $16$ is studied in detail. Orders of the full automorphism groups of all $12$ nonequivalent Vasil'ev codes of length $16$ are found. There are exactly $15$ nonisomorphic systems $S(16,4,3)$ of rank $12$ over $\mathbb{F}_2$, and they can be obtained from codewords of weight $4$ of the extended Vasil'ev codes. Orders of the automorphism groups of all these Steiner systems are found.

Construction of Perfect $q$-ary Codes by Switchings of Simple Components
A. V. Los'
pp. 30–37

Abstract—We suggest a construction of perfect $q$-ary codes by sequential switchings of special-type components (called simple components) of the Hamming code. We prove that such components are minimal. The construction yields a lower bound on the number of different $q$-ary codes; this is presently the best known bound. We show that this bound cannot be substantially improved using switchings of components of this type.

Entropy of Multidimensional Cellular Automata
E. L. Lakshtanov and E. S. Langvagen
pp. 38–45

Abstract—Since the topological entropy of a vast class of two-dimensional cellular automata (CA) is infinite, of interest is the possibility to renormalize it so that to obtain a positive finite value. We find the asymptotics of the information function of a multidimensional CA and, accordingly, introduce the renormalized topological entropy as a coefficient of this asymptotics. We describe some properties of the introduced quantity, in particular, its positivity for CA of the type of  “The Game of Life.” Also, we give an example of an explicit evaluation of this parameter for a particular cellular automaton.

Exact Asymptotics of Large Deviations of Stationary Ornstein–Uhlenbeck Processes for $L^p$-Functionals, $p>0$
V. R. Fatalov
pp. 46–63

Abstract—We prove a general result on the exact asymptotics of the probability $$\operatorname{\bf P} \Biggl \{\int\limits_0^1 |\eta_\gamma(t)|^p\,dt>u^p \Biggr\}$$ as $u \to \infty$, where $p>0$, for a stationary Ornstein–Uhlenbeck process $\eta_\gamma(t)$, i.e., a Gaussian Markov process with zero mean and with the covariance function $\operatorname{\bf E} \eta_\gamma(t) \eta_\gamma(s)=e^{- \gamma |t-s|}$, $t, s\in \mathbb{R}$, $\gamma>0$. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm–Liouville type. For $p=1$ and $p=2$, explicit formulas for the asymptotics are given.

On Raw Coding of Chaotic Dynamics
M. L. Blank
pp. 64–68

Abstract—We study raw coding of trajectories of a chaotic dynamical system by sequences of elements from a finite alphabet and show that there is a fundamental constraint on differences between codes corresponding to different trajectories of the dynamical system.