Abstract
Complete n-tracks in PG ( N , q ) and non-extendable Near MDS codes of dimension N + 1 over F q are known to be equivalent objects. The best known lower bound for the maximum number of points of an n-track is attained by elliptic n-tracks, that is, n-tracks consisting of the F q -rational points of an elliptic curve. This has given a motivation for the study of complete elliptic n-tracks. From previous work, an elliptic n-track in PG ( 2 , q ) is complete provided that either the j-invariant j ( E ) of the underlying elliptic curve E is different from zero, or j ( E ) = 0 and the number N q of F q -rational points of E is even. In this paper it is shown that the latter result extends to odd N q if and only if either q is a square or p ≡ 1 ( mod 3 ) , p being the characteristic of F q . Some completeness results for elliptic n-tracks in dimensions 3 and 5 are also obtained.
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