The thinnest coverings of ellipsoids are studied in the Euclidean spaces of an arbitrary dimension n. Given any ellipsoid, we obtain a tight asymptotic bound on the minimum size of its covering by the balls of radius /spl epsi/. This bound holds for all but the most oblong ellipsoids. The results can be applied to vector quantization when different data streams are bundled together in one block.

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.

It is proved that the Hamming bound is asymptotically tight for codes correcting weakly localized errors, when the number of errors is fixed.

Introduces probabilistic communication models with localized errors and determines 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.>

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.

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.>

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.>