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

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.