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(2,4) all hyperovals are projectively equivalent.
So, for example, with F={0,1,e,e2}, 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 .
Back
Updated 3 October 2006