On Körner-Marton´s sum modulo two problem

In this paper we study the sum modulo two problem proposed by Körner and Marton. In this source coding problem, two transmitters who observe binary sources X and Y, send messages of limited rate to a receiver whose goal is to compute the sum modulo of X and Y. This problem has been solved for the two special cases of independent and symmetric sources. In both of these cases, the rate pair (H(X|Y), H(Y|X)) is achievable. The best known outer bound for this problem is a conventional cut-set bound, and the best known inner bound is derived by Ahlswede and Han using a combination of Slepian-Wolf and Körner-Marton´s coding schemes. In this paper, we propose a new outer bound which is strictly better than the cut-set bound. In particular, we show that the rate pair (H(X|Y), H(Y|X)) is not achievable for any binary sources other than independent and symmetric sources. Then, we study the minimum achievable sum-rate using Ahlswede-Han´s region and propose a conjecture that this amount is not less than minimum of Slepian-Wolf and Körner-Marton´s achievable sum-rates. We provide some evidences for this conjecture.