Completeness of Normal Rational Curves
L. Storme
Seminar of Geometry and Combinatorics, University of Ghent, Krijgslaan 281, B-9000 Ghent, Belgium
DOI: 10.1023/A:1022428405084
Abstract
The completeness of normal rational curves, considered as ( q + 1)-arcs in PG( n, q), is investigated. Previous results of Storme and Thas are improved by using a result by Kovács. This solves the problem completely for large prime numbers q and odd nonsquare prime powers q = p 2 h+1 with p prime, , where p 0( h) is an odd prime number which depends on h.
Pages: 197–202
Keywords: $k$-arcs; normal rational curves; M.D.S. codes
Full Text: PDF
References
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2. H. Kaneta and T. Maruta, "An elementary proof and an extension of Thas' theorem on k-arcs," Math. Proc. Cambridge Philos. Soc. 105 (1989), 459-462.
3. S.J. Kovacs, "Small saturated sets in finite projective planes," Rend. Mat., to appear.
4. G. Seroussi and R.M. Roth, "On M.D.S. extensions of generalized Reed-Solomon codes," IEEE Trans. Inform. Theory IT-32 (1986), 349-354.
5. L. Storme and J.A. Thas, "Generalized Reed-Solomon codes and normal rational curves: an improvement of results by Seroussi and Roth," in Advances in Finite Geometries and Designs, J.W.P. Hirschfeld, D.R. Hughes, and J.A. Thas, eds., Oxford University Press, Oxford, 1991, pp. 369-389.
6. L. Storme and J.A. Thas, "k-arcs and dual k-arcs," Ann. Discrete Math., to appear.
7. J.A. Thas, "Normal rational curves and k-arcs in Galois spaces," Rend. Mat. 1 (1968), 331-334.
8. J.A. Thas, "Connection between the Grassmannian Gk_1;n and the set of the k-arcs of the Galois space Sn,q," Rend. Mat. 2 (1969), 121-134.
9. J.A. Thas, "Projective geometry over a finite field," in Handbook of Geometry, F. Buekenhout, ed, North-Holland, Amsterdam, to appear.
10. J.F. Voloch, "Arcs in projective planes over prime fields," J. Geom. 38 (1990), 198-200.
11. J.F. Voloch, "Complete arcs in Galois planes of non-square order," Advances in Finite Geometries and Designs, J.W.P. Hirschfeld, D.R. Hughes, and J.A. Thas, eds., Oxford University Press, Oxford, 1991, pp. 401-406.