  We consider a binary erasure version of the n-channel multiple descriptions
problem with symmetric descriptions, i.e., the rates of the n descriptions are
the same and the distortion constraint depends only on the number of messages
received. We consider the case where there is no excess rate for every k out of
n descriptions. Our goal is to characterize the achievable distortions D_1,
D_2,...,D_n. We measure the fidelity of reconstruction using two distortion
criteria: an average-case distortion criterion, under which distortion is
measured by taking the average of the per-letter distortion over all source
sequences, and a worst-case distortion criterion, under which distortion is
measured by taking the maximum of the per-letter distortion over all source
sequences. We present achievability schemes, based on random binning for
average-case distortion and systematic MDS (maximum distance separable) codes
for worst-case distortion, and prove optimality results for the corresponding
achievable distortion regions. We then use the binary erasure multiple
descriptions setup to propose a layered coding framework for multiple
descriptions, which we then apply to vector Gaussian multiple descriptions and
prove its optimality for symmetric scalar Gaussian multiple descriptions with
two levels of receivers and no excess rate for the central receiver. We also
prove a new outer bound for the general multi-terminal source coding problem
and use it to prove an optimality result for the robust binary erasure CEO
problem. For the latter, we provide a tight lower bound on the distortion for
\ell messages for any coding scheme that achieves the minimum achievable
distortion for k messages where k is less than or equal to \ell.
