Optimal binary index assignments for a class of equiprobable scalar and vector quantizers

The problem of scalar and vector quantization in conjunction with a noisy binary symmetric channel is considered. The issue is the assignment of the shortest possible distinct binary sequences to quantization levels or vectors so as to minimize the mean-squared error caused by channel errors. By formulating the assignment as a matrix (or vector in the scalar case) and showing that the mean-squared error due to channel errors is determined by the projections of its columns onto the eigenspaces of the multidimensional channel transition matrix, a class of source/quantizer pairs is identified for which the optimal index assignment has a simple and natural form. Among other things, this provides a simpler and more accessible proof of the result of Crimmins et al. (1969) that the natural binary code is an optimal index assignment for the uniform scalar quantizer and uniform source. It also provides a potentially useful approach to further developments in source-channel coding.