Index Assignment for Multichannel Communication under Failure
  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. 
Tanya Y. Berger-Wolf and Edward M. Reingold