Generalized face-uniformity bound for two-valued functions on Hamming graphs

Let $f: \{0,\ldots,q-1\}^n \rightarrow \{0,1\}$ be a function with $B$ zeros and $C$ ones, where $q\geq 3$. Suppose that for some $k$, every $k$-face has the same number

$$\frac{Cq^k}{B+C}$$

of ones of $f$. Generalized face-uniformity bound. If

$$\frac{B+C}{\gcd(B,C)}>q,$$

then

$$k\geq\frac{n}{q+1}+1.$$

This is proposed as a more general form of the perfect-coloring bound, applying to any two-valued function; the case $q=2$ is identified with the known correlation-immunity bound, while the cases $q>2$ remain open.