  We present analytical expressions for optimal entropy-constrained
multiple-description lattice vector quantizers which, under high-resolutions
assumptions, minimize the expected distortion for given packet-loss
probabilities. We consider the asymmetric case where packet-loss probabilities
and side entropies are allowed to be unequal and find optimal quantizers for
any number of descriptions in any dimension. We show that the normalized second
moments of the side-quantizers are given by that of an $L$-dimensional sphere
independent of the choice of lattices. Furthermore, we show that the optimal
bit-distribution among the descriptions is not unique. In fact, within certain
limits, bits can be arbitrarily distributed.
