  We show that extended cyclic codes over $\mathbb{F}_q$ with parameters
$[q+2,3,q]$, $q=2^m$, determine regular hyperovals. We also show that extended
cyclic codes with parameters $[qt-q+t,3,qt-q]$, $1<t<q$, determine (cyclic)
Denniston maximal arcs. Similarly, cyclic codes with parameters
$[q^2+1,4,q^2-q]$ are equivalent to ovoid codes obtained from elliptic quadrics
in $PG(3,q)$. Finally, we give new simple presentations of Denniston maximal
arcs in $PG(2,q)$ and elliptic quadrics in $PG(3,q)$.
