Small complete caps from singular cubics, II

Abstract

Small complete arcs and caps in Galois spaces over finite fields $$\mathbb {F}_q$$ F q with characteristic greater than three are constructed from singular cubic curves. For $$m$$ m a divisor of $$q+1$$ q + 1 or $$q-1$$ q - 1 , complete plane arcs of size approximately $$q/m$$ q / m are obtained, provided that $$(m,6)=1$$ ( m , 6 ) = 1 and $$m<\frac{1}{4}q^{1/4}$$ m < 1 4 q 1 / 4 . If in addition $$m=m_1m_2$$ m = m 1 m 2 with $$(m_1,m_2)=1$$ ( m 1 , m 2 ) = 1 , then complete caps in affine spaces of dimension $$N\equiv 0 \pmod 4$$ N ? 0 ( mod 4 ) with roughly $$\frac{m_1+m_2}{m}q^{N/2}$$ m 1 + m 2 m q N / 2 points are described. These results substantially widen the spectrum of $$q$$ q s for which complete arcs in $$AG(2,q)$$ A G ( 2 , q ) of size approximately $$q^{3/4}$$ q 3 / 4 can be constructed. Complete caps in $$AG(N,q)$$ A G ( N , q ) with roughly $$q^{(4N-1)/8}$$ q ( 4 N - 1 ) / 8 points are also provided. For infinitely many $$q$$ q s, these caps are the smallest known complete caps in $$AG(N,q)$$ A G ( N , q ) , $$N \equiv 0 \pmod 4$$ N ? 0 ( mod 4 ) .