A conic in PG(2,q) has q+1 points, at most two incident with a line.
When q is even all (q+1,2)-arcs can be extended to a (q+2,2), which is called a hyperoval.

So, for example, with F the finite field with 16 elements, the matrix whose columns are {(1,t,t 2) | t ∈ F} ∪ {(0,0,1)} ∪ {(0,1,0)} will generate such a code.

In PG(2,16) they are two projectively inequivalent hyperovals, see [Theorem 8.3.5, J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Second edition, Oxford University Press, Oxford, 1998.]. The regular hyperoval we have seen above and the other is due to [L. Lunelli and M. Sce, k-archi completi nei piani proiettivi desarguesiani di rango 8 e 16. Centro di Calcoli Numerici, Politecnico di Milano, Milano 1958].

The fact that one cannot do better follows from the Griesmer bound .

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Updated 6 October 2006