  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-Shelah lemma and the linear
programming bound. Two lower bounds are given using Gilbert-Varshamov type
arguments over constant-weight and Markov-type sets.
