  We consider multiple description coding for the Gaussian source with K
descriptions under the symmetric mean squared error distortion constraints, and
provide an approximate characterization of the rate region. We show that the
rate region can be sandwiched between two polytopes, between which the gap can
be upper bounded by constants dependent on the number of descriptions, but
independent of the exact distortion constraints. Underlying this result is an
exact characterization of the lossless multi-level diversity source coding
problem: a lossless counterpart of the MD problem. This connection provides a
polytopic template for the inner and outer bounds to the rate region. In order
to establish the outer bound, we generalize Ozarow's technique to introduce a
strategic expansion of the original probability space by more than one random
variables. For the symmetric rate case with any number of descriptions, we show
that the gap between the upper bound and the lower bound for the individual
description rate is no larger than 0.92 bit. The results developed in this work
also suggest the "separation" approach of combining successive refinement
quantization and lossless multi-level diversity coding is a competitive one,
since it is only a constant away from the optimum. The results are further
extended to general sources under the mean squared error distortion measure,
where a similar but looser bound on the gap holds.
