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Small complete arcs and caps in Galois spaces over finite fields $\fq$ withcharacteristic greater than 3 are constructed from cubic curves with anisolated double point. For $m$ a divisor of $q+1$, complete plane arcs of sizeapproximately $q/m$ 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 completecaps of size approximately $\{m_1+m_2}{m}q^{N/2}$ in affine spaces of dimension$N\equiv 0 \pmod 4$ are constructed.

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