Coding for Channels with Localized Errors

Abstract

The following problem was discussed in a lecture by S.I. Gelfand in Oberwolfach (Information Theory, May 1989), based on joint work with L. A. Bassalygo and M. S. Pinsker. We consider a binary channel and we are interested in codes of length n. Let t be given, 0 < t < n. Before a message is transmitted, the sender is given a subset E of cardinality at most t of the positions {1, 2,…, n} in which errors may occur (i.e. outside E all bits are received correctly). The receiver does not know E, but sender and receiver have prearranged codebooks that are used for transmission and reception of M possible messages. The question is to determine F _t (n):= the maximal value of M for which communication over this channel with a code of length n is possible. Results appeared in [2]. We shall give our own proofs of these results and we shall analyze some of the bounds. Furthermore, we discuss a variation in which the subset E has cardinality ≤ t resp. t, and it is known that either all bits in E are received correctly or all of them are incorrect.