Geodesic
Small complete caps from singular cubics. II
Journal of Algebraic Combinatorics, Tome 41 (2015) no. 1, pp. 185-216.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $AG(N,q)$, $N \equiv 0 \pmod 4$.
Classification : 51E20, 51E22
Keywords: Galois affine spaces, bicovering arcs, complete caps, quasi-perfect codes, cubic curves 
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     title = {Small complete caps from singular cubics. {II}},
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Anbar, Nurdagül; Bartoli, Daniele; Platoni, Irene; Giulietti, Massimo. Small complete caps from singular cubics. II. Journal of Algebraic Combinatorics, Tome 41 (2015) no. 1, pp. 185-216. http://geodesic.mathdoc.fr/item/JAC_2015__41_1_a1/