On Aug 19, 7:47*pm, Randy Yates <ya...@ieee.org> asked:
> 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)?

This kind of stuff is routine. Consider, for example, using
a Reed-Solomon code (say (255,223) code over GF(256))
on the binary symmetric (or additive white Gaussian noise)
channel. As another example, one of the first implementations
of a (2,1) convolutional code was for a QPSK modem in which
the two coded bits were the I and Q inputs to the modem,
and the channel rate in bauds was the same as the input
data rate in bits.

>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?

Yes, though the implementation as well as any analysis would
be messy. The first codeword would have its bits in the first
and second QAM symbols transmitted, but the *second*
codeword would have one bit in the second QAM symbol
transmitted, the next four bits in the third QAM symbol, and
two bits in the fourth QAM symbol. The 7 bits of the third
codeword would have .... ; the whole cycle repeating
after 7 QAM symbols (i.e. 4 codewords). Whether the
increased complexity is worth the benefits reaped remains
to be seen in 120 or so pages. Or the answer exists in some
unpublished master's thesis since negative results tend not
to get published: the number of papers saying "Our scheme
works" is far outweighed by the number of papers saying
"We tried this scheme and it did not work."

--Dilip Sarwate