Extended cyclic codes, maximal arcs and ovoids

Abstract

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 \([qt-q+t,3,qt-q]\), \(1<t<q\), q is a power of t, 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 simple presentations of Denniston maximal arcs in PG(2, q) and elliptic quadrics in PG(3, q).