Small complete caps from singular cubics, II

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