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 $${\mathcal{E}}$$ of PG(2, q) via the canonical Veronese embedding$$\nu:\quad (X,Y,Z)\to (X^2,XY,Y^2,XZ,YZ,Z^2).$$(1) If Γ6 (equivalently $${\mathcal{E}}$$) has nGF(q)-rational points, then the associated near-MDS code $${\mathcal{C}}$$ 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 $${\mathcal{C}}$$ 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 $${\mathcal{C}}$$ is also considered.