We investigate the asymptotic rates of length-$n$ binary codes with VC-dimension at most $dn$ and minimum distance at least $delta n$. Two upper bounds are obtained, one as a simple  corollary of a result by Haussler and the other via a shortening approach combining Sauer's lemma and the linear programming bound. Two lower bounds are given using Gilbert-Varshamov type arguments over constant-weight and Markov-type sets respectively.