Small complete arcs and caps in Galois spaces over finite fields $\fq$ with characteristic greater than $3$ 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 $AG(N,q)$, $N \equiv 0 \pmod 4$.
Small complete caps from singular cubics, II
BARTOLI, DANIELE;GIULIETTI, Massimo
2014
Abstract
Small complete arcs and caps in Galois spaces over finite fields $\fq$ with characteristic greater than $3$ 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 $AG(N,q)$, $N \equiv 0 \pmod 4$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
