On the VC-dimension of binary codes

We investigate the asymptotic rates of length-n binary codes with VC-dimension at most dn and minimum distance at least δn. 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.