Leonid A. Bassalygo

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. >

Nonbinary Error-Correcting Codes with One-Time Error-Free Feedback

We show that, using one-time error-free feedback, it is possible to attain the asymptotic Hamming bound if the number of errors is fixed.

A model of restricted asynchronous multiple access in the presence of errors

The paper is based on the brief communication [1], where proofs of results were omitted or drastically abridged. In the present paper, not only this gap is filled but also the very model of restricted asynchronous multiple access from [1] is considered in a somewhat more general situation.

The Grey-Rankin bound for nonbinary codes

The Grey-Rankin bound for nonbinary codes is obtained. Examples of codes meeting this bound are given.

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>>

Correction of Ordinary and Localized Errors

We present upper and lower bounds on the cardinality of a code that corrects both localized and ordinary errors.

Calculation of the Asymptotically Optimal Capacity of a T-User M-Frequency Noiseless Multiple-Access Channel

The statement of the problem is taken from [1]. Let T(T ≥ 2) be the number of users every of which transmits one symbol from the alphabet {1, 2, ..., M}, M ≥ 2, at each time instant (the time is discrete); and the output is a binary sequence of length M where the symbol 0 is in the m-th position if and only if none user transmitted the symbol m. Such channel is referred to as an A-channel in [1]. Denote by X = (X 1, ..., X T ) an M-ary sequence at the input of the channel and by Y = (Y 1, ..., Y M ) a binary sequence at the output. Then (see [1]) the sum capacity of an A-channel is

On maximal number of probability measures on finite set separated in L/sub 1/ metrics

{C_{sum}}(T,M) = \max H(Y),