Non-Gray labeling of error-prone feedback signals

The problem of mapping feedback constellations to bits, to be transmitted over a binary channel subject to errors is considered. In earlier works, it was assumed that using a Gray mapping to map feedback constellations to bits would provide optimal protection against feedback errors. Here it is shown that for 2D feedback constellations with more than 2 bits, and 1D constellations with more than 1 bit, a non-Gray labeling of the constellation is optimal. The reason is that the problem of mapping quantized feedback symbols to bits is a source coding problem, where M-ary symbols are mapped to bits for transmission over a binary channel, not a modulation problem, where bits are mapped to M-ary symbols for transmission over a continuous channel. In concrete terms this means that feedback mode 2 of the 3G WCDMA specifications is slightly suboptimal.