Showing papers in "Problems of Information Transmission in 2006"
TL;DR: The proposed computation method for the discrete Fourier transform is based on factorizing the transform matrix into a product of a binary block circulant matrix and a diagonal block circULant matrix.
Abstract: The discrete Fourier transform over a finite field finds applications in algebraic coding theory. The proposed computation method for the discrete Fourier transform is based on factorizing the transform matrix into a product of a binary block circulant matrix and a diagonal block circulant matrix.
32 citations
TL;DR: In this article, the additivity conjecture holds for a quantum depolarizing channel generated by unitary representations of a discrete group, and it is shown that the additive property of the relative entropy holds for the channel.
Abstract: We consider bistochastic quantum channels generated by unitary representations of a discrete group. We give a proof of the additivity conjecture for a quantum depolarizing channel ? based on the decreasing property of the relative entropy. We show that the additivity conjecture holds for a channel ? = ? o ?, where ? is a phase damping channel.
26 citations
TL;DR: The following generalization to the extension theorem of Hill and Lizak is proved: for every nonextendable linear [n, k, d]q code, q = ps, (d,q) = 1, where q + r(q) + 1 is the smallest size of a nontrivial blocking set in PG(2, q).
Abstract: We prove the following generalization to the extension theorem of Hill and Lizak: For every nonextendable linear [n, k, d] q code, q = p s , (d,q) = 1, we have $$\sum\limits_{i
ot \equiv 0,d(\bmod q)} {A_i > q^{k - 3} r(q),} $$ where q + r(q) + 1 is the smallest size of a nontrivial blocking set in PG(2, q). This result is applied further to rule out the existence of some linear codes over $$\mathbb{F}_4 $$ meeting the Griesmer bound.
15 citations
TL;DR: In this paper, a new approach for upper bounding the channel reliability function using the code spectrum is described, which allows to treat both the low and high rate cases in a unified way.
Abstract: A new approach for upper bounding the channel reliability function using the code spectrum is described. It allows to treat both the low and high rate cases in a unified way. In particular, previously known upper bounds are improved, and a new derivation of the sphere-packing bound is presented.
15 citations
TL;DR: It is proved that among allNonequivalent extended binary perfect (16, 4, 211) codes there are exactly 1719 nonequivalent codes of rank 14 over F2.
Abstract: All extended binary perfect (16, 4, 211) codes of rank 14 over the field F 2 are classified. It is proved that among all nonequivalent extended binary perfect (16, 4, 211) codes there are exactly 1719 nonequivalent codes of rank 14 over F 2. Among these codes there are 844 codes classified by Phelps (Solov'eva-Phelps codes) and 875 other codes obtained by the construction of Etzion-Vardy and by a new general doubling construction, presented in the paper. Thus, the only open question in the classification of extended binary perfect (16,4,211) codes is that on such codes of rank 15 over F 2.
13 citations
TL;DR: It is shown that the bound on the number of different q-ary codes cannot be substantially improved using switchings of components of this type, and it is proved that such components are minimal.
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.
12 citations
TL;DR: The Laplace method is used for Gaussian measures in Banach spaces and 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.
Abstract: We prove a general result on the exact asymptotics of the probability $$P\left\{ {\int\limits_0^1 {\left| {\eta _\gamma (t)} \right|^p dt > u^p } } \right\}$$ as u ? ?, where p > 0, for a stationary Ornstein-Uhlenbeck process ? ?(t), i.e., a Gaussian Markov process with zero mean and with the covariance function E??(t)??(s), t, s ? ?, ? > 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.
11 citations
TL;DR: It is proved that the stochastic process describing the configuration of a particle system weakly converges in the fluid-dynamic limit to a deterministic process, which is a solution of a certain partial differential equation.
Abstract: We study limit dynamics of a system of interacting particles, which is one of possible models for the parallel and distributed computation process. For a rather wide class of multi-particle interactions, we prove that the stochastic process describing the configuration of a particle system weakly converges in the fluid-dynamic limit to a deterministic process, which is a solution of a certain partial differential equation.
10 citations
TL;DR: A new concept of modified Butson-Hadamard matrices is introduced and two families of quaternary codes derived from the corresponding families of modified complex matrices with entries from a finite cyclic group of order 4 are constructed.
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.
8 citations
TL;DR: A construction and examples of linear two-dimensional array codes correcting rectangular burst errors of size b1 × b2 with minimum redundancy r = 2b1b2 and constructions of cyclicTwo-dimensionalarray codes correcting phased and arbitrary rectangular Burst errors are presented.
Abstract: Two-dimensional array codes correcting rectangular burst errors are considered. We give a construction and examples of linear two-dimensional array codes correcting rectangular burst errors of size b 1 × b 2 with minimum redundancy r = 2b 1 b 2. We present constructions of cyclic two-dimensional array codes correcting phased and arbitrary rectangular burst errors; their encoding and decoding algorithms are also given. A class of cyclic two-dimensional array codes correcting rectangular burst errors with asymptotically minimal redundancy is described. We construct a class of linear two-dimensional array codes correcting cyclic rectangular b 1 × b 2 burst errors with asymptotic excess redundancy $$\tilde r_C (b_1 ,b_2 ) = 2b_1 b_2 - 3$$ .
8 citations
TL;DR: The Grey-Rankin bound for nonbinary codes is obtained and examples of codes meeting this bound are given.
Abstract: The Grey-Rankin bound for nonbinary codes is obtained Examples of codes meeting this bound are given
TL;DR: It is shown that strong additivity of the Holevo capacity for two given channels provides projective relations between optimal set for the tensor product of these channels and optimal sets for the individual channels.
Abstract: Special sets of states, called optimal, which are related to the Holevo capacity and to the minimal output entropy of a quantum channel, are considered. By methods of convex analysis and operator theory, structural properties of optimal sets and conditions of their coincidence are explored for an arbitrary channel. It is shown that strong additivity of the Holevo capacity for two given channels provides projective relations between optimal sets for the tensor product of these channels and optimal sets for the individual channels.
TL;DR: This paper considers a Gaussian multiple-input multiple-output (MIMO) discrete-time vector channel with memory and shows what the dependence of input signal components should be to give this capacity.
Abstract: The paper considers a Gaussian multiple-input multiple-output (MIMO) discrete-time vector channel with memory. The problem is to find the capacity of such a channel. It is known that the capacity of Gaussian vector channels with memory was given in [1]. In the present paper, we show a different approach, which uses another definition of the capacity. For a channel with n = 2 inputs and outputs, this approach gives an expression for the capacity which is different from that in [1]. The paper shows what the dependence of input signal components should be to give this capacity. A multidimensional water-filling interpretation works for the optimum vector input signal power distribution but cannot work for the description of the input component dependences. For the case of n ? 3 inputs and outputs, we give a lower bound on the channel capacity.
TL;DR: An upper bound on the error probability (first error event) of product convolutional codes over a memoryless binary symmetric channel, and the resulting error exponent are derived and it is shown that, for both decoding methods, the error probabilities can be attained with nonexponentially increasing decoding complexity.
Abstract: An upper bound on the error probability (first error event) of product convolutional codes over a memoryless binary symmetric channel, and the resulting error exponent are derived. The error exponent is estimated for two decoding procedures. It is shown that, for both decoding methods, the error probability exponentially decreasing with the constraint length of product convolutional codes can be attained with nonexponentially increasing decoding complexity. Both estimated error exponents are similar to those for woven convolutional codes with outer and inner warp.
TL;DR: An averaging property for a one-server queuing process is established, M(t)/G/1, crucial in the study of the Poisson hypothesis and its implications include the statement that the output flow always possesses more regularity than the input flow.
Abstract: We establish an averaging property for a one-server queuing process, M(t)/G/1. It is a new relation between the output flow rate and the input flow rate, crucial in the study of the Poisson hypothesis. Its implications include the statement that the output flow always possesses more regularity than the input flow.
TL;DR: The renormalized topological entropy is introduced as a coefficient of the asymptotics of the information function of a multidimensional CA and some properties of the introduced quantity are described, in particular, its positivity for CA of the type of “The Game of Life.
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.
TL;DR: The goal of the republication of this article is not only to acquaint the reader with a scarcely known article of the great scientist but to invite him to observe the process of scientific progress.
Abstract: Kolmogorov's article "Contemporary Debates on the Nature of Mathematics" was published in 1929 in Nauchnoe slovo journal (no. 6, pp. 41---54) and has not been republished since then. At the end of the 19th and at the beginning of the 20th century, abstractions of a very high order appeared and were rooted in mathematics, so that their correlation with reality became a crucial question. A number of great mathematicians of the time were interested in this question, and Kolmogorov's article was quite up to date. The goal of the republication of this article, which is currently difficult to access, is not only to acquaint the reader with a scarcely known article of the great scientist but to invite him to observe the process of scientific progress. It should be noted that, only a year after the publication of the article, some astonishing results of Godel appeared that gave answers to a number of questions in Kolmogorov's article and brought mathematical strictness in the very formulation of these questions. Therefore, the article is accompanied by external comments in order to give an insight into the current state of the art. In the text the comments are indicated by numbers in brackets, from [1] to [21].1 As for the rest, Kolmogorov's text is published in the same way as it was published in Nauchnoe slovo, without any alterations of the sense. Accompanying comments given to the article by the editors of Nauchnoe slovo (footnotes and introduction to the article) are omitted.
TL;DR: All 708 103 nonisomorphic Steiner systems S(16, 4, 3, 3) of order 16 and rank 14 over F2 are enumerated and it is shown that all of them can be obtained by the general doubling construction presented in the paper.
Abstract: All 708 103 nonisomorphic Steiner systems S(16, 4, 3) of order 16 and rank 14 over F 2 are enumerated. Among them there are exactly 1059 homogeneous systems. It is shown that all the 708 103 Steiner systems can be obtained by the general doubling construction presented in the paper.
TL;DR: A lower bound for the number of iteratively correctable erasures is given, with application to the ensemble of LDPC codes with parity-check matrices composed of permutation matrices, which is O(N log N), where N is the block length.
Abstract: A lower bound for the number of iteratively correctable erasures is given, with application to the ensemble of LDPC codes with parity-check matrices composed of permutation matrices [1] We assume that the Zyablov-Pinsker iterative decoding algorithm [2] is used Its complexity is O(N log N), where N is the block length
TL;DR: There is a fundamental constraint on differences between codes corresponding to different trajectories of a chaotic dynamical system by sequences of elements from a finite alphabet.
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.
TL;DR: Two new families of nonlinear q-ary codes derived from the corresponding families of modified Butson Hadamard matrices are constructed, with very easy construction and decoding procedures.
Abstract: In this paper we construct two new families of nonlinear q-ary codes derived from the corresponding families of modified Butson Hadamard matrices. These codes have very easy construction and decoding procedures, and their parameters are rather close to the Plotkin bound.
TL;DR: Two new families of nonlinear senary codes derived from the corresponding families of modified Butson-Hadamard matrices are constructed, and their parameters are very close to the Plotkin bound.
Abstract: In this paper we construct two new families of nonlinear senary codes derived from the corresponding families of modified Butson-Hadamard matrices. These codes have easy construction and decoding procedures, and their parameters are very close to the Plotkin bound.
TL;DR: It is proved that any Steiner system S(n = 2m, 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.
Abstract: Extended binary perfect nonlinear Vasil'ev codes of length n = 2m and Steiner systems S(n, 4, 3) of rank n-m over 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 F 2. We prove that any Steiner system S(n = 2m, 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 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.
TL;DR: This work studies the reliability exponent of the additive exponential noise channel (AENC) in the absence as well as in the presence of noiseless feedback to obtain a lower bound on the random-transmission-time reliability exponent.
Abstract: We study the reliability exponent of the additive exponential noise channel (AENC) in the absence as well as in the presence of noiseless feedback. For rates above the critical rate, the fixed-transmission-time reliability exponent of the AENC is completely determined, while below the critical rate an expurgated exponent is obtained. Using a fixed-block-length code ensemble (with block length denoting the number of recorded departures), we obtain a lower bound on the random-transmission-time reliability exponent of the AENC. Finally, with a variable-block-length code ensemble, a lower bound on the random-transmission-time zero-error capacity of the AENC with noiseless feedback is obtained.
TL;DR: Product-form formulas for the computation of stationary distributions of the considered networks are obtained, and algorithms for the solution of auxiliary problems are developed.
Abstract: The paper is devoted to the computation of stationary distributions of queueing systems in random media. Results obtained for considered models follow from a theorem proved in the paper. As an application of the theorem, we consider Jackson networks whose structure (the set of working nodes, service and input flow intensities, routing matrix, state set) and type (open/closed network) varies as the state of another network or of the environment changes. Product-form formulas for the computation of stationary distributions of the considered networks are obtained, and algorithms for the solution of auxiliary problems are developed.
TL;DR: The results, which are obtained in terms of algorithmic complexity, allow to discuss heuristically the properties of sufficiently complex security portfolios in the conditions of daily changing return rates.
Abstract: For a simplest mathematical model of a stock market, the problem of optimal distribution of investments among different securities (stocks, bonds, etc.) is considered. Our results, which are obtained in terms of algorithmic complexity, allow to discuss heuristically the properties of sufficiently complex security portfolios in the conditions of daily changing return rates. All considerations are given in the combinatorial framework and do not use any probabilistic models.
TL;DR: The precise value of the function FN(m, n, k), which is the number of binary words of length N and weight m that contain an arbitrary word of length n and weight k as a fragment, is found.
Abstract: We find a precise value of the function F N(m, n, k), which is the number of binary words of length N and weight m that contain an arbitrary word of length n and weight k as a fragment. As a consequence, we obtain a known result on the number of binary words of length N that contain a fixed word of length n as a fragment.
TL;DR: It is proved that the stationary distribution of the network states is invariant with respect to a functional form of workload distributions if the first moments are fixed.
Abstract: We consider open and closed preemptive-resume queueing systems with absolute priority of incoming customers. Single-server nodes have several service modes (regimes); the time of switching between the modes is exponential. Switching can be made to adjacent modes only. The amount of work required for servicing an incoming customer (workload) is a random variable with an arbitrary distribution function. For an open network, the input flow is Poissonian. We prove that the stationary distribution of the network states is invariant with respect to a functional form of workload distributions if the first moments are fixed.
TL;DR: Maximal extensions of Post classes containing 0, 1, and x in the algebra of partially unreliable Boolean functions are described and criteria of expressibility of Boolean functions by circuits in a basis of partially reliable elements are proved.
Abstract: Maximal extensions of Post classes containing 0, 1, and x in the algebra of partially unreliable Boolean functions are described. Based on these extensions, criteria of expressibility of Boolean functions by circuits in a basis of partially unreliable elements are proved.
TL;DR: It is proved that such rates are optimal in a minimax sense for the signal S in a given class of smooth functions, and the convergence rate for the estimation risks is obtained.
Abstract: The paper concerns the estimation of a smooth signal S(t) and its derivatives in the presence of a noise depending on a small parameter ? based on a partial observation. A nonlinear Kalman-type filter is proposed to perform on-line estimation. For the signal S in a given class of smooth functions, the convergence rate for the estimation risks, as ? ? 0, is obtained. It is proved that such rates are optimal in a minimax sense. In contrast to the complete observation case, the rates are reduced, due to incomplete information.