New results in binary multiple descriptions

An encoder whose input is a binary equiprobable memoryless source produces one output of rate R_{1} and another of rate R_{2} . Let D_{1}, D_{2}, and D_{0} , respectively, denote the average error frequencies with which the source data can be reproduced on the basis of the encoder output of rate R_{l} only, the encoder output of rate R_{2} only, and both encoder outputs. The two-descriptions problem is to determine the region R of all quintuples (R_{1}, R_{2}, D_{1}, D_{2}, D_{0}) that are achievable in thc usual Shannon sense. Let R(D)=1+D \log_{2} D+(1-D) \log_{2}(1-D) denote the error frequency rate-distortion function of the source. The "no excess rate case" prevails when R_{1} + R_{2} = R(D_{0}) , and the "excess rate case" when R_{1} + R_{2} > R(D_{0}) . Denote the section of R at (R_{1}, R_{2}, D_{0}) by D(R_{1} R_{2}, D_{0}) =\{(D_{1},D_{2}): (R_{1}, R_{2}, D_{1},D_{2},D_{0}) \in R} . In the no excess rate case we show that a portion of the boundary of D(R_{1}, R_{2}, D_{0}) coincides with the curve (\frac{1}{2} + D_{1}-2D_{0})(\frac_{1}_{2} + D_{2}-2D_{0})= \frac{1}{2}(1-2D_{0})^{2} . This curve is an extension of Witsenhausen's hyperbola bound to the case D_{0} > 0 . It follows that the projection of R onto the (D_{1}, D_{2}) -plane at fixed D_{0} consists of all D_{1} \geq D_{0} and D_{2} \geq D_{0} that lie on or above this hyperbola. In the excess rate case we show by counterexample that the achievable region of El Gamal and Cover is not tight.