  In this paper we derive analytical expressions for the central and side
quantizers which, under high-resolutions assumptions, minimize the expected
distortion of a symmetric multiple-description lattice vector quantization
(MD-LVQ) system subject to entropy constraints on the side descriptions for
given packet-loss probabilities.
  We consider a special case of the general n-channel symmetric
multiple-description problem where only a single parameter controls the
redundancy tradeoffs between the central and the side distortions. Previous
work on two-channel MD-LVQ showed that the distortions of the side quantizers
can be expressed through the normalized second moment of a sphere. We show here
that this is also the case for three-channel MD-LVQ. Furthermore, we conjecture
that this is true for the general n-channel MD-LVQ.
  For given source, target rate and packet-loss probabilities we find the
optimal number of descriptions and construct the MD-LVQ system that minimizes
the expected distortion. We verify theoretical expressions by numerical
simulations and show in a practical setup that significant performance
improvements can be achieved over state-of-the-art two-channel MD-LVQ by using
three-channel MD-LVQ.
