Complete arcs in PG (2, 25): The spectrum of the sizes and the classification of the smallest complete arcs

Discrete Mathematics

Volumn 307, Issue 6, 2007, Pages 739-747

  Marcugini, Stefano ^a   Milani, Alfredo ^a   Pambianco, Fernanda ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

Author keywords

Arcs; Complete arcs; Projective planes

Indexed keywords

NUMERICAL METHODS; SPECTRUM ANALYSIS;

AUTOMORPHISM; COMPLETE ARCS; COMPUTER SEARCH; PROJECTIVE PLANES;

COMPUTATIONAL GEOMETRY;

EID: 33846047585     PISSN: 0012365X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.disc.2005.11.094     Document Type: Article

References (32)

1. A.H. Ali, Classification of arcs in Galois plane of order thirteen, Ph.D. Thesis, University of Sussex, 1993.
2. S. Ball, On sets of points in finite planes, Ph.D. Thesis, University of Sussex, 1994.
3. Blokhuis A. Note on the size of a blocking set in PG (2, p). Combinatorica 14 (1994) 111-114
4. Bose R.C. Mathematical theory of the symmetrical factorial design. Sankhya 8 (1947) 107-166
5. Chao J.M., and Kaneta H. Classical arcs in PG (r, q) for 23 ≤ q ≤ 29. Discrete Math. 226 (2001) 377-385
6. 2). J. Geom. 53 (1995) 37-40
7. Addendum, J. Geom. 59 (1997) 32-33.
8. Faina G., Marcugini S., Milani A., and Pambianco F. The spectrum of the values k for which there exists a complete k-arc in PG (2, q) for q ≤ 23. Ars Combin. 47 (1997) 3-11
9. Faina G., and Pambianco F. On the spectrum of values k for which a complete k-cap in PG (n, q) exists. J. Geom. 62 (1998) 84-98
10. Gordon C.E. Orbits of arcs in PG (N, K) under projectivities. Geom. Dedicata 42 (1992) 187-203
11. Hirschfeld J.W.P. Projective Geometries Over Finite Fields (1998), Clarendon Press, Oxford
12. Hirschfeld J.W.P., and Korchmáros G. Arcs and curves over a finite field. Finite Field Appl. 5 4 (1999) 393-408
13. Hirschfeld J.W.P., and Sadeh A. The projective plane over the field of eleven elements. Mitt. Math. Sem. Giessen 164 (1984) 245-257
14. Hirschfeld J.W.P., and Storme L. The packing problem in statistics, coding theory and finite projective spaces: update 2001. In: Blokhuis A., Hirschfeld J.W.P., Jungnickel D., and Thas J.A. (Eds). Developments in Mathematics, Finite Geometries vol. 3 (2000), Dordrecht, Kluwer, The Netherlands 201-246
15. Kenstenband B.C. Unitals intersections in finite projective planes. Geom. Dedicata 11 (1981) 107-117
16. Korchmáros G. Osservazioni sui risultati di B. Segre relativi ai k-archi completi contenenti k - 1 punti di una conica. Atti Accad. Naz. Lincei Rend. 56 (1974) 541-549
17. Lam C.W.H., Kolesova G., and Thiel L. A computer search for finite projective planes of order 9. Discrete Math. 92 (1991) 187-195
18. P. Lisonek, Computer-assisted studies in algebraic combinatorics, Ph.D. Thesis, Research Institute for Symbolic Computation, J. Kepler University of Linz, RISC-Linz Report Series No. 94-68, 1994.
19. 2, q. Boll. Un. Mat. Ital. 11 (1956) 178-181
20. S. Marcugini, A. Milani, F. Pambianco, A computer search for complete arcs in PG (2, q), q ≤ 128, Rapporto tecnico n. 18/95, Università degli Studi di Perugia, 1995.
21. S. Marcugini, A. Milani, F. Pambianco, A computer search for orbits of complete arcs in PG (2, q), q = 8, 13, 16, 23, Rapporto tecnico n. 10/98, Università degli Studi di Perugia, 1998.
22. S. Marcugini, A. Milani, F. Pambianco, Complete arcs in PG (2, 25): the spectrum of the sizes and the list of the smallest complete arcs, Rapporto tecnico n. 6/98, Università degli Studi di Perugia, 1998.
23. Marcugini S., Milani A., and Pambianco F. Minimal complete arcs in PG (2, q), q ≤ 29. J. Combin. Math. Combin. Comput. 47 (2003) 19-29
24. T. Penttila, G.F. Royle, Private Communication, 1995.
25. Referee's remark.
26. Royle G.F. An orderly algorithm and some applications to finite geometry. Discrete Math. 185 1-3 (1998) 105-115
27. Segre B. Ovals in a finite projective plane. Canad. J. Math. 7 (1955) 414-416
28. Segre B. Curve razionali normali e k-archi negli spazi finiti. Ann. Mat. Pura Appl. 39 (1955) 357-379
29. Segre B. Le geometrie di Galois. Ann. Mat. Pura Appl. 48 (1959) 1-97
30. Szo{combining double acute accent}nyi T. Arcs, caps, codes and 3-independent subsets. In: Faina G., and Tallini G. (Eds). Proceedings of the International Conference Giornate di Geometrie Combinatorie Perugia 1992 (1993), Università degli studi di Perugia 57-80
31. Tallini G. Le geometrie di Galois e le loro applicazioni alla statistica e alla teoria delle informazioni. Rend. Mat. e Appl. 19 (1960) 379-400
32. J.A. Thas, M.D.S. codes and arcs in projective spaces. A survey, Le matematiche (1992) 315-328.