  A cap set is a subset of $\mathbb{F}_3^n$ with no solutions to $x+y+z=0$
other than when $x=y=z$. In this paper, we provide a new lower bound on the
size of a maximal cap set. Building on a construction of Edel, we use improved
computational methods and new theoretical ideas to show that, for large enough
$n$, there is always a cap set in $\mathbb{F}_3^n$ of size at least $2.218^n$.
