@JohnWatrous Ah, of course! You are correct. I've made another thoughtless false assumption "$\langle{\phi\vert{A_+}\vert\phi}\rangle > 0$ implies $\langle{\phi\vert{A_-}\vert\phi}\rangle = 0$."

@am567 There should be $2^4$ basis vectors for the quantum code since each $x \in \mathbb{F}_2^4$ ($x$ not limited to just a basis of vectors of $\mathbb{F}_2^4$) corresponds to a code basis vector $| \psi_{x} \rangle$.

I guess what I said about FT over Fq/FT on Fq can be nitpicking because the terms seem to be interchanged in the literature, but my point was that studying a "QFT over Fq" should be concerned with replacing groups with a finite field, rather than replacing the field C with a different ring. Basically, the Wikipedia article is a different concept.

I think this is a sort of the point you make in the 3rd paragraph, but if you are interested in a QFT, I don't think a FT over Fq is what you are interested in (i.e. considering Fq-valued functions on some group). But rather, a FT on Fq (i.e. C-valued functions on Fq). Here are sources: sites.math.rutgers.edu/~sk1233/courses/finitefields-F13/… (pp. 5) people.cs.uchicago.edu/~laci/reu02/fourier.pdf (pp. 16). Basically, the additive and multiplicative groups of Fq are considered in tandem. The QFT in question seems to only consider the additive part (see 1st source).