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 3description 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.