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I'm not sure I can find an explicit formula. Wondering if anyone can come up with lower/upper bounds.

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The exact answer is unknown in general. One standard upper bound for $q-$ary codes is the Singleton bound, which gives $$|C|\leq q^{n-d+1},$$ and codes meeting this bound are called MDS.

A lower bound is the Gilbert-Varshamov bound, given by

$$|C|\geq \frac{q^n}{\mathrm{Vol}_q(d-1)}=\frac{q^n}{\sum_{k=0}^{d-1} \binom{n}{k}(q-1)^k}.$$ Using these keywords should get you started, there are many more results.

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If you're not looking for asymptotic results, there are extensive tables that are maintained by researchers.

You can find them at www.codetables.de. Go to that webpage, and click "linear codes". They have complete tables of known upper and lower bounds for linear GF(3) (ternary) codes with $n \leq 243$.

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