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?
>

I *believe* ( but could easily be wrong ) that the map
from symbol states to symbol order sets is opportunistic,
and based on which sorts of maps empirically behave the best.

Mathematically, any function could work. And whether or
not the function is bijective/symmetrical seems
irrelevant, long as you don't need the symmetry for
computational reasons.

ObDisclosure: I'm not a coding theory guy at all, but
I've worked with several on instrumentation.

> 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?

I don't see why not - isn't the choice of
16 QAM largely independent of what's being modulated
over it? Once you have conditions necessary to
describe one bit in 16QAM... it's a bit. It is,
form a dataflow perspective, just a by-16 PSK stream....
once you get it into baseband...

16 & 7 modularize together in an interesting, 56-bit
sort of way, too....

--
Les Cargill