Complete arcs and complete caps from cubics with an isolated double point

TL;DR

This paper constructs small complete arcs and caps in finite projective and affine spaces using cubic curves with isolated double points, providing new explicit geometric configurations with sizes related to divisors of q+1.

Contribution

It introduces novel constructions of complete arcs and caps from cubic curves with isolated double points, expanding the known configurations in finite geometry.

Findings

Complete arcs of size ~q/m for divisors m of q+1

Complete caps of size ~((m_1 + m_2)/m) q^{N/2} in affine spaces

Conditions on m for the constructions to hold

Abstract

Small complete arcs and caps in Galois spaces over finite fields $\fq$ with characteristic greater than 3 are constructed from cubic curves with an isolated double point. For $m$ a divisor of $q+1$, complete plane arcs of size approximately $q/m$ are obtained, provided that $(m,6)=1$ and $m<\{1}{4}q^{1/4}$. If in addition $m=m_1m_2$ with $(m_1,m_2)=1$, then complete caps of size approximately $\{m_1+m_2}{m}q^{N/2}$ in affine spaces of dimension $N\equiv 0 \pmod 4$ are constructed.