There are three projectively distinct examples of a (9,3)-arcs in PG(2,4) and so three inequivalent [9,3,6]-linear codes over the finite field with 4 elements.
1. A hermitian curve in PG(2,q) has q√q+1 points, at most √q+1 incident with a line.
All hermitian curves are projectively equivalent. So, for example, the matrix whose columns are distinct points of PG(2,4) that are zeros of the polynomial x√q+1+y√q+1+z√q+1 will generate such a code.
2.
For example, with e2=e+1, the code with generator matrix
| 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | e | 1+e | e | e | 1+e | 1+e |
| 0 | e | e+1 | 1+e | e | 1 | 0 | 0 | 1 |
3.
The complement of three non-concurrent lines. For example, the matrix whose columns are distinct points of PG(2,4) with at least two non-zero coordinates.
This classification is due to L. Lunelli and M. Sce [Considerazioni arithmetiche e resultati sperimentali sui {K;n}q archi, Ist.
Lombardo Accad. Sci. Rend. A, 98 (1964), 3--52]. Although I used to have this article, I cannot find it now and so I cannot verify if it was already known that there is no (10,3)-arc.
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Updated 9 October 2006