Small complete caps from nodal cubics

Abstract

Bicovering arcs in Galois affine planes of odd order are a powerful tool for constructing complete caps in spaces of higher dimensions. In this paper we investigate whether some arcs contained in nodal cubic curves are bicovering. For $m_1$, $m_2$ coprime divisors of $q-1$, bicovering arcs in $AG(2,q)$ of size $k\le (q-1)\frac{m_1+m_2}{m_1{m}_2}$ are obtained, provided that $(m_1{m}_2,6)=1$ and $m_1{m}_2<\sqrt[4]{q}/3.5$. Such arcs produce complete caps of size $k{q}^{(N-2)/2}$ in affine spaces of dimension $N\equiv 0(\mathrm{mod}4)$. For infinitely many $q$'s these caps are the smallest known complete caps in $AG(N,q)$, $N\equiv 0(\mathrm{mod}4)$.

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