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.
In PG(2,8) all hyperovals are projectively equivalent, see [Corollary 8.32, J.W.P. Hirschfeld, Projective Geometries over Finite
Fields, Second edition, Oxford University Press, Oxford, 1998.]
So, for example, with F={0,1,e,1+e,e2,1+e2,e+e2,1+e+e2} where e3=e+1, the matrix whose columns are {(1,t,t 2) | t ∈ F} ∪ {(0,0,1)} ∪ {(0,1,0)} will generate such a code.
The fact that one cannot do better follows from the Griesmer bound .
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Updated 6 October 2006