Small complete caps from singular cubics, II

Small complete arcs and caps in Galois spaces over finite fields $$\mathbb {F}_q$$Fq 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<14q1/4. If in addition $$m=m_1m_2$$m=m1m2 with $$(m_1,m_2)=1$$(m1,m2)=1, then complete caps in affine spaces of dimension $$N\equiv 0 \pmod 4$$N≡0(mod4) with roughly $$\frac{m_1+m_2}{m}q^{N/2}$$m1+m2mqN/2 points are described. These results substantially widen the spectrum of $$q$$qs for which complete arcs in $$AG(2,q)$$AG(2,q) of size approximately $$q^{3/4}$$q3/4 can be constructed. Complete caps in $$AG(N,q)$$AG(N,q) with roughly $$q^{(4N-1)/8}$$q(4N-1)/8 points are also provided. For infinitely many $$q$$qs, these caps are the smallest known complete caps in $$AG(N,q)$$AG(N,q), $$N \equiv 0 \pmod 4$$N≡0(mod4).