Asymmetric multi-level diversity coding

Symmetric multilevel diversity coding was introduced by Roche et al, where a set of K information sources is encoded by K encoders and the decoders reconstruct sources 1, . . . , k, where k is the number of encoders to which they have access. In this paper, we formulate an asymmetric multilevel diversity coding problem, where a set of 2(K)-1 information sources is encoded by K encoders into K streams/descriptions. There are 2(K)-1 decoders, each of which has access to a non-empty subset of the encoded messages. The decoders are assigned with ordered levels, and each of them has to decode a subset of the information sources, according to its level, which depends on the set of encoders to which it has access, not just the cardinality. We obtain a single letter characterization of the complete achievable rate region for the 3-description problem. In doing so, we show that it is necessary to jointly encode independent sources (i.e., similar to network coding), and that linear codes are optimal for this problem.