Elliptic near-MDS codes over $${\mathbb{F}}_5$$

Let Γ₆ be the elliptic curve of degree 6 in PG(5, q) arising from a non-singular cubic curve of PG(2, q) via the canonical Veronese embedding(1) If Γ₆ (equivalently ) has n GF(q)-rational points, then the associated near-MDS code 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 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 is also considered.