Quantizers with uniform encoders and channel optimized decoders

Scalar quantizers with uniform encoders and channel optimized decoders are studied for uniform sources and binary symmetric channels. It is shown that the natural binary code (NBC) and folded binary code (FBC) induce point density functions that are uniform on proper subintervals of the source support, whereas the Gray code (GC) does not induce a point density function. The mean-squared errors (MSE) for the NBC, FBC, GC, and for randomly chosen index assignments are calculated and the NBC is shown to be mean-squared optimal among all possible index assignments, for all bit-error rates and all quantizer transmission rates. In contrast, it is shown that almost all index assignments perform poorly and have degenerate codebooks.