Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-19T06:47:51.906Z Has data issue: false hasContentIssue false
  • English
  • Français

An elementary proof and an extension of Thas' theorem on k-arcs

Published online by Cambridge University Press:  04 October 2011

Hitoshi Kaneta  and
Tatsuya Maruta
Hitoshi Kaneta
Affiliation:
Department of Mathematics, Okayama University, Okayama 700, Japan
Tatsuya Maruta
Affiliation:
Department of Mathematics, Okayama University, Okayama 700, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

Let q be the finite field of q elements. Denote by Sr q the projective space of dimension r over q. In Sr,q, where r ≥ 2, a k-arc is defined (see [4]) as a set of k points such that no j + 2 lie in a Sj,q, for j = 1,2,…, r−1. (For a k-arc with k > r, this last condition holds for all j when it holds for j = r−1.) A rational curve Cn of order n in Sr,q, is the set

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 459 - 462
Copyright
Copyright © Cambridge Philosophical Society 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Casse, L. R. A. and Glynn, D. G.. On the uniqueness of (q + 1)4-arcs of PG (4, q), q = 2n, h ≥ 3. Discrete Math. 48 (1984), 173–186.CrossRefGoogle Scholar
[2] Hirschfeld, J. W. P.. Projective Geometries over Finite Fields (Oxford University Press, 1979).Google Scholar
[3] Hirschfeld, J. W. P.. Finite Projective Space of Three Dimensions (Oxford University Press, 1985).Google Scholar
[4] Segre, B.. Curve razionali normali e k-archi negli spazi finiti. Ann. Mat. Pura. Appl. 39 (1955), 357–379.CrossRefGoogle Scholar
[5] Segre, B.. Introduction to Galois geometries. Atti. Accad. Naz. Lincei Mem. 8 (1967), 133–236.Google Scholar
[6] Thas, J. A.. Normal rational curves and k-arcs in Galois spaces. Rend. Mat. 1 (1968), 331–334.Google Scholar
[7] Thas, J. A.. Complete arcs and algebraic curves in PG(2,q). J. Algebra 106 (1987), 451–464.CrossRefGoogle Scholar
[8] Walker, R. T.. Algebraic Curves (Princeton University Press, 1950).Google Scholar