Small complete caps from singular cubics, II
Small complete arcs and caps in Galois spaces over finite fields \(\mathbb {F}_q\) 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 \(m<\frac{1}{4}q^{1/4}\). If in addition \(m=m_1m_2\) with \((m_1,m_2)=1\), then complete caps in affine spaces of dimension \(N\equiv 0 \pmod 4\) with roughly \(\frac{m_1+m_2}{m}q^{N/2}\) points are described. These results substantially widen the spectrum of \(q\)s 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 \(q\)s, these caps are the smallest known complete caps in \(AG(N,q)\), \(N \equiv 0 \pmod 4\).
complete caps cubic curves galois affine spaces bicovering arcs quasi-perfect codes 51e20