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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
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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

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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