  This paper investigates the problem of finding an optimal nonbinary index
assignment from (M) quantization levels of a maximum entropy scalar quantizer
to (M)-PSK symbols transmitted over a symmetric memoryless channel with
additive noise following decreasing probability density function (such as the
AWGN channel) so as to minimize the channel mean-squared distortion. The
so-called zigzag mapping under maximum-likelihood (ML) decoding was known to be
asymptotically optimal, but the problem of determining the optimal index
assignment for any given signal-to-noise ratio (SNR) is still open. Based on a
generalized version of the Hardy-Littlewood convolution-rearrangement
inequality, we prove that the zigzag mapping under ML decoding is optimal for
all SNRs. It is further proved that the same optimality results also hold under
minimum mean-square-error (MMSE) decoding. Numerical results are presented to
verify our optimality results and to demonstrate the performance gain of the
optimal (M)-ary index assignment over the state-of-the-art binary counterpart
for the case of (8)-PSK over the AWGN channel.
