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

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Abstract

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 n GF(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.

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Correspondence to Vito Abatangelo.

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Communicated by P. Wild.

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Abatangelo, V., Larato, B. Elliptic near-MDS codes over \({\mathbb{F}}_5\) . Des. Codes Cryptogr. 46, 167–174 (2008). https://doi.org/10.1007/s10623-007-9144-8

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  • DOI: https://doi.org/10.1007/s10623-007-9144-8

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