A conic in PG(2,q) has q+1 points, at most two incident with a line.
All conics are projectively equivalent and all (q+1,2)-arcs are conics when q is odd; this is Segre's theorem.
So, for example, with F={0,1,2,3,4}, the matrix whose columns are {(1,t,t 2) | t ∈ F} ∪ {(0,0,1)} will generate such a code.
The fact that one cannot do better is due to Bose [R.C. Bose, Mathematical theory of the symmetrical factorial design,
Sankhya 8 (1947), 107--166.]
The Griesmer bound gives n ≤ q+2 and then one has to show that n=q+2 implies that q is even.
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Updated 4 October 2006