Mark S. Pinsker

Sensitivity of channel capacity

In some channels subject to crosstalk or other types of additive interference, the noise is the sum of a dominant Gaussian noise and a relatively weak non Gaussian contaminating noise. Although the capacity of such channels cannot be evaluated in general, the authors analyze the decrease in capacity, or sensitivity of the channel capacity to the weak contaminating noise. The main result is that for a very large class of contaminating noise processes, explicit expressions for the sensitivity of a discrete-time channel capacity do exist. Moreover, in those cases the sensitivity depends on the contaminating process distribution only through its autocorrelation function and so it coincides with the sensitivity with respect to a Gaussian contaminating noise with the same autocorrelation function.

Nonbinary codes correcting localized errors

A recursive construction of asymptotically dense codes correcting a constant number of localized errors is examined. The codes overcome difficulties for a particular case with an asymptotic Hamming bound in which the number of errors increases linearly with the length of codes. It is shown that this method is applicable to both binary and nonbinary cases. >

Coding for partially localized errors

A model of a communication system over the binary channel is studied. The authors consider the situation when the encoder knows the positions where the errors can (but not necessarily will) occur. In other words, the encoder knows that all symbols outside these dubious positions are transmitted without errors, while those on dubious positions may suffer from errors. On the other hand, the decoder does not have any prior information about the possible error locations. This model, the channel with localized errors, is believed to be useful in analyzing various data storage problems. >

On Estimation of Information via Variation

New inequalities relating mutual information and variational distance are derived.

On coverings of ellipsoids in Euclidean spaces

The thinnest coverings of ellipsoids are studied in the Euclidean spaces of an arbitrary dimension n. Given any ellipsoid, the main goal is to find its /spl epsiv/-entropy, which is the logarithm of the minimum number of the balls of radius /spl epsiv/ needed to cover this ellipsoid. A tight asymptotic bound on the /spl epsiv/-entropy is obtained for all but the most oblong ellipsoids, which have very high eccentricity. This bound depends only on the volume of the sub-ellipsoid spanned over all the axes of the original ellipsoid, whose length (diameter) exceeds 2/spl epsiv/. The results can be applied to vector quantization performed when data streams from different sources are bundled together in one block.

Epsilon-entropy of an Ellipsoid in a Hamming Space

Asymptotic behavior of the ε-entropy of an ellipsoid in a Hamming space is investigated as the dimension of the space grows.

On the thinnest coverings of spheres and ellipsoids with balls in hamming and euclidean spaces

In this paper, we present some new results on the thinnest coverings that can be obtained in Hamming or Euclidean spaces if spheres and ellipsoids are covered with balls of some radius e. In particular, we tighten the bounds currently known for the e-entropy of Hamming spheres of an arbitrary radius r. New bounds for the e-entropy of Hamming balls are also derived. If both parameters e and r are linear in dimension n, then the upper bounds exceed the lower ones by an additive term of order logn. We also present the uniform bounds valid for all values of e and r. In the second part of the paper, new sufficient conditions are obtained, which allow one to verify the validity of the asymptotic formula for the size of an ellipsoid in a Hamming space. Finally, we survey recent results concerning coverings of ellipsoids in Hamming and Euclidean spaces.

Sensitivity of Gaussian channel capacity and rate-distortion function to nonGaussian contamination

In some applications, channel noise is the sum of a Gaussian noise and a relatively weak non-Gaussian contaminating noise. Although the capacity of such channels cannot be evaluated in general, we analyze the decrease in capacity, or sensitivity of the channel capacity to the weak contaminating noise. We show that for a very large class of contaminating noise processes, explicit expressions for the sensitivity of a discrete-time channel capacity do exist. Sensitivity is shown to depend on the contaminating process distribution only through its autocorrelation function and so it coincides with the sensitivity with respect to a Gaussian contaminating noise with the same autocorrelation function. A key result is a formula for the derivative of the water-filling capacity with respect to the contaminating noise power. Parallel results for the sensitivity of rate-distortion function relative to a mean-square-error criterion of almost Gaussian processes are obtained.

An Optimization Problem Related to the Computation of the Epsilon-entropy of an Ellipsoid in a Hamming Space

The problem of optimizing (finding the maximin of) the difference between the entropy functions of two n-dimensional vectors under special restrictions on their components is solved. This optimum gives the main term of the asymptotics for the ε-entropy of an ellipsoid in a Hamming space as the dimension of the space grows.

Localized random and arbitrary errors in the light of AV channel theory

A central problem in coding theory consists in finding bounds for the maximal size, say N(n,2t+1,q), of a t-error correcting code over a q-ary alphabet with blocklength n. This code concept is suited for communication over a q-ary channel with input and output alphabet X={0,1,...,q-1}, when a word of length n sent by the encoder is changed by the channel in at most t letters. Neither the encoder nor the decoder knows in advance where the errors, that is changes of letters, occur. Bassalygo, Gelfand, and Pinsker introduced the concept of localized errors. They assume that the encoder, who wants to encode message m, knows the t-element set E/spl sub/[n]={1,2,...,n} of positions, in which errors may occur. The encoder can make the codeword, representing m, dependent on E/spl isin//spl Escr//sub t/, the family of t-elements subsets of [n]. The authors call them a priori error pattern. The set of associated (a posteriori) errors is V(E)={e/sup n/=(e/sub 1/,...,e/sub n/)/spl isin/X/sup n/:e/sub t/=0 for t/spl notin/E}. They introduce probabilistic communication models with localized errors and determine the optimal rates of codes, if a priori error patterns or actual errors or both occur at random according to uniform distributions. There are strong connections to the theory of arbitrarily varying channels. The authors also have new coding technique for additive arbitrarily varying channels.<<ETX>>