Algebraic capsets

TL;DR

This paper introduces new algebraic constructions of capsets in finite fields, achieving the smallest known complete capsets with sizes close to theoretical lower bounds.

Contribution

It presents novel algebraic methods to construct minimal complete capsets in _3^n, improving upon previous size bounds.

Findings

Constructed the smallest known complete capsets.

Achieved sizes proportional to the best known lower bounds.

Demonstrated algebraic equations over field extensions are effective for capset construction.

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.