On the completeness of certain n-tracks arising from elliptic curves

Complete n-tracks in PG(N,q) and non-extendable Near MDS codes of dimension N+1 over Fq 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 Fq-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 Nq of Fq-rational points of E is even. In this paper it is shown that the latter result extends to odd Nq if and only if either q is a square or , p being the characteristic of Fq. Some completeness results for elliptic n-tracks in dimensions 3 and 5 are also obtained.