Randy Yates wrote:
> Does a modulation with q different symbol states absolutely require a
> code over X^n, where X = {a_1, a_2, ..., a_q}? Or can you use a
> symbol set (X) with a different order (count)?
> A slightly different question: is this *typically* done (q symbol states
> ==> order q symbol set)? I.e., although it's not required, is it
> typically done this way?

Ideally you want coding to be matched to the modulation, so the Hamming
distance corresponds to the Euclidean distance.

> As a simple example, could a binary (7,4) Hamming code be applied to 16
> QAM, such that 4/7 codewords fit into one symbol?

The result is likely to be worse then the uncoded QAM. Proakis Chapter 8
is exactly about your questions.

Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com