Extended cyclic codes, maximal arcs and ovoids

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