Admissibility criterion for perfect 2-colorings of Hamming graphs

For fixed parameters $q$, $b$, and $c$, call $(q,b,c)$ admissible if there is an integer $n_0=n_0(b,c;q)$ such that a $(b,c)$-coloring of $H(n,q)$ exists if and only if $n\geq n_0$. Assume $b\geq c\ne 1$. Admissibility criterion. The parameters $q$, $b$, and $c$ are admissible if and only if

$$\frac{b+c}{\gcd(b,c)}\mid q^k$$

for some $k\in\mathbb{N}$. The preceding results provide necessary conditions and establish the threshold formulation; the stated equivalence is put forward on the basis of those results and computations, and remains open.