Bound covering constants for larger alphabets

Determine whether C_q is finite for every fixed integer q≥3 and whether the constants C_q can be bounded by an absolute constant independent of q, where C_q=sup_{n≥1} f_q(n)/(q/(q−1))^n and f_q(n) is the minimum number of sets Q_a^{(q)} covering Z_q^n.

Background

For q≥2, the paper defines Q_a^{(q)}={x∈Z_q^n:x_i≠a_i for every coordinate i} and lets f_q(n) be the minimum number of such sets needed to cover Z_q^n. The cases q=2 and q=3 are respectively trivial and the principal subject of the paper.

The fractional-covering lower bound is (q/(q−1))^n, while the Johnson–Lovász–Stein estimate gives an upper bound with an additional factor of order n. The authors define C_q as the supremum of the normalized covering numbers and ask whether this quantity is finite for each fixed q and uniformly bounded over all q.