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

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• 作者：Massimo Giulietti ; Fabio Pasticci
• 关键词：Projective spaces ; n-Tracks ; Near MDS codes ; Elliptic curves
• 刊名：Finite Fields and Their Applications
• 出版年：2007
• 出版时间：November 2007
• 年：2007
• 卷：13
• 期：4
• 页码：988-1000
• 全文大小：169 K

文摘

Complete n-tracks in CDTF-2&_mathId=mml1&_user=10&_cdi=6798&_rdoc=24&_acct=C000050221&_version=1&_userid=10&md5=9680ba189690dd4e49773f7ee8e667da"" title=""Click to view the MathML source"">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 of the underlying elliptic curve is different from zero, or and the number N_q of F_q-rational points of 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 being the characteristic of F_q. Some completeness results for elliptic n-tracks in dimensions 3 and 5 are also obtained.