The listening-helper source-coding problem for a doubly symmetric binary source

In the listening-helper source-coding problem Encoder 1 and Decoder 1 observe respectively a pair of i.i.d. correlated sources and both wish to communicate the first source to Decoder 2 subject to a distortion constraint. Encoder 1 sends a message to both decoders and then Decoder 1 - the listening-helper - sends a message just to Decoder 2. For the case of a doubly binary symmetric source and Hamming distortion we determine the rate region for this problem as a function of the distortion constraint. In the derivation of the outer bound we show that the evaluation may be restricted to a set of auxiliary random variables satisfying a long Markov structure.