On multiple descriptions and team guessing

Witsenhausen's hyperbola bound for the multiple description problem without excess rate in case of a binary source is not right for exact joint reproductions. However, this bound is tight for almost{exact joint reproductions (Theorem 1, conjectured by Witsenhausen). The proof is based on an approximative form of the team guessing lemma for sequences of random variables. (This result may be of interest also for team guessing.) The hyperbola bound is also tight for exact joint reproductions and arbitrarily small, but possible, excess rate (Theorem 2). The proof of this result uses our covering lemma. 1 The Problem of Multiple Descriptions During the last years a strong interest has developed in a certain source{coding problem called the \problem of multiple descriptions". Since the origin of this problem and the motivations for its study have already been extensively described (see 1]{{9]), we begin immediately with the formal setup. Let (X t) 1 t=1 be a sequence of independent and identically distributed (i.i.d.) random variables (RV's) with values in a nite set X, that is, a discrete memoryless source (DMS). We are given three nite reconstruction spaces, ^ For a function F deened on a product space Y n we use the notation rate(F) = 1 n log kFk; kFk = the cardinality of the range of F: