Is the statement true or false? If $d \leq q-1$, then $B_q(q-1,d) = q^{q-d}$
It is clear from the singleton bound that $B_q(q-1,d) \leq q^{q-d}$, however I am having trouble trying to find the existence of this code. I know that if this code exist then it would be a MDS code. I even tried using Gibert-Varshamov's theorem, however to proof it seem like I have to go through a mess of $$V_q^n(r) = \sum_{s=0}^r \binom{n}{s}(q-1)^s.$$ Furthermore, I tried to prove that the statement maybe false however, most of my examples only deal with binary which holds true. Much help is appreciated, or point me to a similar question.