Extended cyclic codes, maximal arcs and ovoids

We show that extended cyclic codes over 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 [q²+ 1 , 4 , q²- 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).