# On the VC-Dimension of Binary Codes

## TL;DR

This paper explores the limits of binary codes with constrained VC-dimension and minimum distance, providing new upper and lower bounds on their asymptotic rates.

## Contribution

It introduces novel upper bounds derived from existing theorems and new lower bounds using Gilbert-Varshamov type arguments for codes with VC-dimension constraints.

## Key findings

- Two upper bounds on asymptotic code rates with VC-dimension constraints
- Two lower bounds based on Gilbert-Varshamov type constructions
- Insights into the trade-offs between VC-dimension, code length, and minimum distance

## Abstract

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.