  We consider the problem of multiple description scalar quantizers and
describing the achievable rate-distortion tuples in that setting. We formulate
it as a combinatorial optimization problem of arranging numbers in a matrix to
minimize the maximum difference between the largest and the smallest number in
any row or column. We develop a technique for deriving lower bounds on the
distortion at given channel rates. The approach is constructive, thus allowing
an algorithm that gives a closely matching upper bound. For the case of two
communication channels with equal rates, the bounds coincide, thus giving the
precise lowest achievable distortion at fixed rates. The bounds are within a
small constant for higher number of channels. To the best of our knowledge,
this is the first result concerning systems with more than two communication
channels. The problem is also equivalent to the bandwidth minimization problem
of Hamming graphs.
