Cyclic Arcs in PG(2, q)
L. Storme
and H. Van Maldeghem
DOI: 10.1023/A:1022454221497
Abstract
B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete ( q 2 - q + l)-arcs in PG(2, q 2), q 
3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q 2 - q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.
3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q 2 - q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.Pages: 113–128
Keywords: $k$-arc; conic; M.D.S. code; cyclic group
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References
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2. P. Dembowski, Finite Geometries, Springer-Verlag, New York, 1968.
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6. J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Oxford University Press, Oxford, England, 1991.
7. D.R. Hughes and F. Piper, Projective Planes, Springer-Verlag, Berlin, 1973.
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11. CM. O'Keefe and T. Penttila, "A new hyperoval in PG(2, 32)," J. Geom. 44 (1992), 117-139.
12. T. Penttila and I. Pinneri, "Irregular hyperovals in PG(2, 64)," J. Geom, to appear.
13. G. Pickert, "Free mobility in affine planes," Proc. of the Conference on Algebra and Geometry, Kuwait, (1981), 153-157.
14. A.R. Sadeh, "Classification of k-arcs in PG(2, 11) and cubic surfaces in PG(3, 11)," Ph. D. thesis, University of Sussex, England,
1984. STORME AND VAN MALDEGHEM
15. B. Segre, Lectures on Modem Geometry. Cremonese, Roma, 1961.
16. B. Segre, "Ovali e curve a nei piani di Galois di caratteristica due," Atti dell' Accad. Naz. Lined Rend. 32 (8) (1962), 785-790.
2. P. Dembowski, Finite Geometries, Springer-Verlag, New York, 1968.
3. J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas, "Complete arcs in planes of square order," Ann. Discrete Math. 30 (1986), 243-250.
4. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Oxford University Press, Oxford, England, 1979.
5. J.W.P Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford, England, 1985.
6. J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Oxford University Press, Oxford, England, 1991.
7. D.R. Hughes and F. Piper, Projective Planes, Springer-Verlag, Berlin, 1973.
8. B. Huppert, Endliche Gruppen I, Springer, Berlin, 1967.
9. B.C. Kestenband, "A family of complete arcs in finite projective planes," Colloq. Math LVII (1989), 59-67.
10. L. Lombardo-Radice, "Sul problema dei k-archi completi di S2,q," Boll. Un. Mat. Ital. 11 (1956), 178-181.
11. CM. O'Keefe and T. Penttila, "A new hyperoval in PG(2, 32)," J. Geom. 44 (1992), 117-139.
12. T. Penttila and I. Pinneri, "Irregular hyperovals in PG(2, 64)," J. Geom, to appear.
13. G. Pickert, "Free mobility in affine planes," Proc. of the Conference on Algebra and Geometry, Kuwait, (1981), 153-157.
14. A.R. Sadeh, "Classification of k-arcs in PG(2, 11) and cubic surfaces in PG(3, 11)," Ph. D. thesis, University of Sussex, England,
1984. STORME AND VAN MALDEGHEM
15. B. Segre, Lectures on Modem Geometry. Cremonese, Roma, 1961.
16. B. Segre, "Ovali e curve a nei piani di Galois di caratteristica due," Atti dell' Accad. Naz. Lined Rend. 32 (8) (1962), 785-790.