Algebraic capsets

Abstract

Capsets are subsets of ${\mathbb{F}}_3^n$ with no three points on a line and a capset is complete if it is not a subset of a larger capset. We study some new constructions of capsets via algebraic equations over extensions of ${\mathbb{F}}_3$. In particular we construct the smallest known complete capsets with size proportional to the best known lower bound.