Elliptic near-MDS codes over F-5
Let Gamma(6) be the elliptic curve of degree 6 in PG(5, q) arising from a non-singular cubic curve epsilon of PG(2, q) via the canonical Veronese embedding nu: (X, Y, Z) -> (X-2, XY, Y-2, XZ, YZ, Z(2)). (1) If Gamma(6) (equivalently epsilon) has n GF(q)-rational points, then the associated near- MDS code C has length n and dimension 6. In this paper, the case q = 5 is investigated. For q = 5, the maximum number of GF (q)-rational points of an elliptic curve is known to be equal to ten. We show that for an elliptic curve with ten GF (5)-rational points, the associated near-MDS code C can be extended by adding two more points of PG(5, 5). In this way we obtain six non- isomorphic [12, 6](5) codes. The automorphism group of C is also considered.
codes elliptic curves finite fields