Showing papers in "Problems of Information Transmission in 2010"
TL;DR: In this article, the authors studied the properties of quantum mutual information and coherent information in the infinite-dimensional case, and their properties were studied in detail, and an upper bound for the coherent information was obtained.
Abstract: The paper is devoted to the study of quantum mutual information and coherent information, two important characteristics of a quantum communication channel. Appropriate definitions of these quantities in the infinite-dimensional case are given, and their properties are studied in detail. Basic identities relating the quantum mutual information and coherent information of a pair of complementary channels are proved. An unexpected continuity property of the quantum mutual information and coherent information, following from the above identities, is observed. An upper bound for the coherent information is obtained.
34 citations
TL;DR: A new set of necessary and sufficient conditions at the transition points, which offer new insights into the transition and make the computation of the optimal distribution easier.
Abstract: The capacity-achieving input distribution for many channels like the additive white Gaussian noise (AWGN) channel and the free-space optical intensity (FSOI) channel under the peak-power constraint is discrete with a finite number of mass points The number of mass points is itself a variable, and figuring it out is a part of the optimization problem We wish to understand the behavior of the optimal input distribution at the transition points where the number of mass points changes To this end, we give a new set of necessary and sufficient conditions at the transition points, which offer new insights into the transition and make the computation of the optimal distribution easier For the real AWGN channel case, we show that for the zero-mean unit-variance Gaussian noise, the peak amplitude A of 1671 and 2786 mark the points where the binary and ternary signaling, respectively, are no longer optimal For the FSOI channel, we give transition points where binary gives way to ternary, and in some cases where ternary gives way to quaternary, in the presence of the peak-power constraint and with or without the average-power constraint
33 citations
TL;DR: This work considers the decoding for Silva-Kschischang-Kötter random network codes based on Gabidulin’s rank-metric codes and presents an algorithm for simultaneous correction of rank errors and generalized erasures.
Abstract: We consider the decoding for Silva-Kschischang-Kotter random network codes based on Gabidulin's rank-metric codes. The model of a random network coding channel can be reduced to transmitting matrices of a rank code through a channel introducing three types of additive errors. The first type is called random rank errors. To describe other types, the notions of generalized row erasures and generalized column erasures are introduced. An algorithm for simultaneous correction of rank errors and generalized erasures is presented. An example is given.
23 citations
TL;DR: It is shown that if the crossover probability of the feedback channel is less than a certain positive value, then the achievable error exponent is better than the decoding error exponent of a channel without feedback.
Abstract: A binary symmetric channel is used for information transmission. There is also another noisy binary symmetric channel (feedback channel), and the transmitter observes without delay all outputs of the forward channel via the feedback channel. Transmission of an exponential number of messages is considered (i.e., the transmission rate is positive). The achievable decoding error exponent for this combination of channels is studied. It is shown that if the crossover probability of the feedback channel is less than a certain positive value, then the achievable error exponent is better than the decoding error exponent of a channel without feedback.
16 citations
TL;DR: It is shown that under this decoding algorithm there are codes in the ensemble with the number of correctable errors linearly growing with the code length, and a condition on the vertex expansion of the Tanner graph corresponding to the code is weakened.
Abstract: We consider an ensemble of random q-ary LDPC codes. As constituent codes, we use q-ary single-parity-check codes with d = 2 and Reed-Solomon codes with d = 3. We propose a hard-decision iterative decoding algorithm with the number of iterations of the order of the logarithm of the code length. We show that under this decoding algorithm there are codes in the ensemble with the number of correctable errors linearly growing with the code length. We weaken a condition on the vertex expansion of the Tanner graph corresponding to the code.
16 citations
TL;DR: A new group testing model is proposed, which is related to separating codes and cover-free codes, and it is proposed that this model should be applied to QR codes.
Abstract: We propose a new group testing model, which is related to separating codes and cover-free codes.
14 citations
TL;DR: This paper shows that a random (conditional on \bar a $$) oracle b does not help to extract common information from the strings ai, and investigates the problem for a special class of properties (for properties that can be expressed by an ∃-formula).
Abstract: Assume that a tuple of binary strings $$ \bar a $$ = ?a 1 ..., a n ? has negligible mutual information with another string b. Does this mean that properties of the Kolmogorov complexity of $$ \bar a $$ do not change significantly if we relativize them to b? This question becomes very nontrivial when we try to formalize it. In this paper we investigate this problem for a special class of properties (for properties that can be expressed by an ?-formula). In particular, we show that a random (conditional on $$ \bar a $$ ) oracle b does not help to extract common information from the strings a i .
14 citations
TL;DR: Derivation of the results is based on the method of comparing with a Wiener process and numerical values of the asymptotics in the case p = 1, p = 2, and for the sup-norm are presented.
Abstract: Let w(t) be a standard Wiener process, w(0) = 0, and let ηa(t) = w(t + a) - w(t), t ≥ 0, be increments of the Wiener process, a > 0. Let Z a (t), t [0, 2a], be a zeromean Gaussian stationary a.s. continuous process with a covariance function of the form EZ a (t)Z a (s) = 1/2[a - |t - s|], t, s - [0, 2a]. For 0 0 of the probabilities $$ P\left\{ {\int\limits_0^T {\left| {\eta _a \left( t \right)} \right|^p dt \leqslant \varepsilon ^p } } \right\} for T \leqslant a, P\left\{ {\int\limits_0^T {\left| {Z_a \left( t \right)} \right|^p dt \leqslant \varepsilon ^p } } \right\} for T < 2a $$, and compute similar asymptotics for the sup-norm. Derivation of the results is based on the method of comparing with a Wiener process. We present numerical values of the asymptotics in the case p = 1, p = 2, and for the sup-norm. We also consider application of the obtained results to one functional quantization problem of information theory.
10 citations
TL;DR: The notion of a discrete Walsh function is refined and generalization is generalized, for which a method for generating a corresponding W-matrix is proposed for the discrete Walsh transform of the Paley enumeration.
Abstract: We refine the notion of a discrete Walsh function and generalize the notion of a discrete Walsh transform, for which we propose a method for generating a corresponding W-matrix We propose spectral decompositions of the discrete Walsh transform operators in arbitrary enumerations, as well as methods for finding bases of eigenspaces, one of them using a new direct product of matrices We propose a notation for the fast discrete Walsh transform algorithm in the Paley enumeration We construct Parseval frames for eigenspaces of the discrete Walsh transform in the Paley enumeration and demonstrate methods for applying them in error detection and correction
10 citations
TL;DR: This work considers the case where the outer decoder is able to decode beyond half the minimum distance of the outer code, and derives achievable decoding radii for GMD decoding.
Abstract: For decoding concatenated codes up to half their designed distance, generalized minimum distance (GMD) decoding can be used. GMD decoding applies multitrial error/erasure decoding of the outer code, where erased symbols depend on some reliability measure stemming from the inner decoders. We consider the case where the outer decoder is able to decode beyond half the minimum distance of the outer code. For a given number of outer decoding trials, we derive achievable decoding radii for GMD decoding. Vice versa, we give a lower bound on the number of required outer decoding trials to obtain the greatest possible decoding radius.
9 citations
Journal Article•
TL;DR: In this article, the achievable decoding error exponent for such a combination of channels is investigated, and it is shown that if the crossover probability of the feedback channel is less than a certain positive value, then the achievable error exponent is better than the decoding error of the channel without feedback.
Abstract: For information transmission a binary symmetric channel is used. There is also another noisy binary symmetric channel (feedback channel), and the transmitter observes without delay all the outputs of the forward channel via that feedback channel. The transmission of an exponential number of messages (i.e. the transmission rate is positive) is considered. The achievable decoding error exponent for such a combination of channels is investigated. It is shown that if the crossover probability of the feedback channel is less than a certain positive value, then the achievable error exponent is better than the decoding error exponent of the channel without feedback.
TL;DR: The discrete Chrestenson-Kronecker transform is a linear transform whose matrix is a Kronecker power of the matrix of the discrete Fourier transform with respect to a new direct product of matrices.
Abstract: The discrete Chrestenson-Kronecker transform is a linear transform whose matrix is a Kronecker power of the matrix of the discrete Fourier transform. The matrix of the discrete Chrestenson-Levy transform is represented as a power of the matrix of the discrete Fourier transform with respect to a new direct product of matrices. We study properties of and analyze fast algorithms for these two main kinds of the discrete Chrestenson transform. We consider properties of and construction methods for other types of the discrete Chrestenson transform.
TL;DR: To study a mathematical model of a random access network with a finite number of sources, retrials, and a conflict warning stage, a method of asymptotic semiinvariants under a growing number of Sources is proposed, which allows for numerical implementation of a prelimit distribution of the number of requests in a retrial pool.
Abstract: To study a mathematical model of a random access network with a finite number of sources, retrials, and a conflict warning stage, we propose a method of asymptotic semiinvariants under a growing number of sources, which allows us to find the asymptotic probability distribution of the number of requests in a retrial pool. We present results of numerical implementation of a prelimit distribution of the number of requests in the retrial pool. We compare the prelimit and asymptotic semiinvariants.
TL;DR: It is found that asymptotic mean-square errors of semirecursive nonparametric estimators of functionals of a multidimensional density function under the assumption that observations satisfy a strong mixing condition are found.
Abstract: We find principal parts of asymptotic mean-square errors of semirecursive nonparametric estimators of functionals of a multidimensional density function under the assumption that observations satisfy a strong mixing condition. Results are illustrated by an example of a nonlinear autoregression process.
TL;DR: Under rather general conditions, results on sharp asymptotics of the probabilities of n → ∞ are proved, namely, the Laplace method for sojourn times of discrete-time Markov chains.
Abstract: Let {? k } k=0 ? be a sequence of i.i.d. real-valued random variables, and let g(x) be a continuous positive function. Under rather general conditions, we prove results on sharp asymptotics of the probabilities $$ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) 0, and exponential random variables with g(x) = x for x ? 0.
TL;DR: It is established sufficient conditions for polynomial rate of convergence to a stationary distribution and of beta-mixing in continuous-time Erlang-type systems.
Abstract: We establish sufficient conditions for polynomial rate of convergence to a stationary distribution and of beta-mixing in continuous-time Erlang-type systems. Our results are a natural complement both to results of Erlang himself, dating back to the beginning of the 20th century, and to exponential estimates established later.
TL;DR: An explicit formula is obtained which in a special case allows us to calculate the maximum of mutual information of several random variables via the variational distance between the joint distribution of these random variables and the product of their marginal distributions.
Abstract: This paper supplements the author's paper [1]. We obtain an explicit formula which in a special case allows us to calculate the maximum of mutual information of several random variables via the variational distance between the joint distribution of these random variables and the product of their marginal distributions. We establish two new inequalities for the binary entropy function, which are related to the problem considered here.
TL;DR: Codes with words that have no identical symbols are defined, their relation to permutation codes is observed, and an optimization problem for them is state.
Abstract: We consider sequences in which every symbol of an alphabet occurs at most once. We construct families of such sequences as nonlinear subcodes of a q-ary [n, k, n ? k + 1] q Reed-Solomon code of length n ? q consisting of words that have no identical symbols. We introduce the notion of a bunch of words of a linear code. For dimensions k ? 3 we obtain constructive lower estimates (tight bounds in a number of cases) on the maximum cardinality of a subcode for various n and q, and construct subsets of words meeting these estimates and bounds. We define codes with words that have no identical symbols, observe their relation to permutation codes, and state an optimization problem for them.
TL;DR: It is shown that in the partially ordered set of P2-degrees there are no maximal elements and it is proved that above each incomplete P1-degree there is a continuum of P3- Degrees.
Abstract: We study the partially ordered set of Boolean P 2-degrees. We introduce the notions of complete and incomplete Boolean degrees. We show that for each complete P 2-degree there exist both a countable decreasing chain of P 2-degrees and a countable antichain of P 2-degrees. We prove that above each incomplete P 2-degree there is a continuum of P 2-degrees. Thus, in total we show that in the partially ordered set of P 2-degrees there are no maximal elements.
TL;DR: This work considers pseudorandom sequences v over a field GF(pr) obtained by mapping ℓ-grams of a linear recurring sequence u over a Galois ring to an arbitrary coordinate set and studies the possibility of uniquely reconstructing u given v.
Abstract: We consider pseudorandom sequences v over a field GF(p r ) obtained by mapping ?-grams of a linear recurring sequence u over a Galois ring to an arbitrary coordinate set We study the possibility of uniquely reconstructing u given v Earlier known results are briefly overviewed
TL;DR: The authors complete deriving a formula for the reliability function of a binary symmetric channel (BSC) for list decoding in the case of zero rate.
Abstract: We complete deriving a formula for the reliability function of a binary symmetric channel (BSC) for list decoding in the case of zero rate.
TL;DR: It is proved that arbitrary n-ary quasigroups of order 4 can be transformed into each other by successive switchings of {a, b}-components.
Abstract: We prove that arbitrary n-ary quasigroups of order 4 can be transformed into each other by successive switchings of {a, b}-components. We prove that perfect (closely packed) binary codes with distance 3 whose rank (dimension of the linear span) is greater by 1 or 2 than the rank of a linear perfect code can be taken to each other by successive switchings of i-components.
Journal Article•
TL;DR: A new model for road detection, based on the multiple birth and death (MBD) optimization algorithm, recently proposed for counting populations of trees or flamingos is introduced.
Abstract: In this paper, we introduce a new model for road detection, based on the multiple birth and death (MBD) optimization algorithm, recently proposed for counting populations of trees or flamingos. We discuss the choice of the energy function guaranteeing accurate detections of both junctions and parallel roads, and preserving the connectivity of the road network despite the occlusions in high resolution images. We also propose here a novel local filter used as a preprocessing step, for detecting roads, and involved in the definition of the energy function.
TL;DR: This work considers a decentralized multiple access system with a binary “success-failure“ feedback and introduces a family of algorithms (protocols) called “algorithms with delayed intervals“ and study stability conditions of one of them.
Abstract: We consider a decentralized multiple access system with a binary "success-failure" feedback. We introduce a family of algorithms (protocols) called "algorithms with delayed intervals" and study stability conditions of one of them. Then we discuss some numerical results and a number of related and interesting problems and hypotheses.
TL;DR: An algorithm for enumeration and denumeration of words with given constraints on run lengths of ones (dklr-sequences) is proposed, where n is the length of enumerated words, whereas for the best known methods it is at least cn, c > 0.
Abstract: We propose an algorithm for enumeration and denumeration of words with given constraints on run lengths of ones (dklr-sequences). For large n, operation time of the algorithm (per symbol of a sequence) is at most O(log3 n log log n), where n is the length of enumerated words, whereas for the best known methods it is at least cn, c > 0.
TL;DR: The present paper derives explicit formulas for sphere sizes in this metric and considers their asymptotics in the case of spheres of a constant radius and obtains a random coding bound and Hamming bound for the optimal size of the so-called DNA codes for the case for a constant distance.
Abstract: For any two q-ary sequences x and y, the stem similarity between them is defined as a total number of stems (blocks of length 2 consisting of adjacent elements of x and y) in their longest common Hamming subsequence. For q = 4 this similarity function and the corresponding distance function arise in molecular biology in describing an additive mathematical model of thermodynamic distance between DNA sequences. In the present paper, we derive explicit formulas for sphere sizes in this metric and consider their asymptotics in the case of spheres of a constant radius. Based on these results, we also obtain a random coding bound and Hamming bound for the optimal size of the so-called DNA codes for the case of a constant distance.
TL;DR: An improved bound for the dimension is obtained and an exact value of the minimum distance is found for the chain of equivalent and embedded codes of q-ary separable Goppa codes.
Abstract: We show that subclasses of q-ary separable Goppa codes Γ(L, G) with L = {? ? GF(q 2?): G(?) ? 0} and with special Goppa polynomials G(x) can be represented as a chain of equivalent and embedded codes. For all codes of the chain we obtain an improved bound for the dimension and find an exact value of the minimum distance. A chain of binary codes is considered as a particular case with specific properties.
TL;DR: An explicit construction of all extended perfect codes of length 16 and rank 13 with a given kernel size is presented and perfect binary codes of lengths 15 and 16 and of rank 13 obtained by the general doubling construction are classified.
Abstract: Inaccuracies in computations in the paper of the authors on classification of perfect binary codes of lengths 15 and 16 and of rank 13 are fixed. An explicit construction of all extended perfect codes of length 16 and rank 13 with a given kernel size is presented. Perfect binary codes of length 15 and rank 14 obtained by the general doubling construction are classified.
TL;DR: This work proposes a method for computing the number of q-ary sequences that contain the subblock 00 precisely r times.
Abstract: We propose a method for computing the number of q-ary sequences that contain the subblock 00 precisely r times.