On Berger-Tung Inner Bound for Sum-Rate versus Sum-Distortion Problem

While tightness of the Berger-Tung inner bound has been established in the quadratic Gaussian case, and its slackness has been demonstrated in another case dealing with sources with common information, the underlying tightness/slackness issue re-mains to be settled in several scenarios. In this context, seeking to study a simple variant of the Berger-Tung problem, we consider doubly symmetric binary sources, Hamming distortion measures, and sum-rate versus sum-distortion. As a first step, in this paper we propose two functions admitting closed-form expressions, prove their local optimality in certain sense, conjecture that those functions specify the Berger-Tung inner bound, and present simulation-based evidence in support of such conjecture.