On the spectrum of minimal blocking sets in PG(2, q)

Journal of Geometry

Volumn 76, Issue 1-2, 2003, Pages 256-281

  Szonyi, Tamás ^a,b   Gács, András ^a   Weiner, Zsuzsa ^a,b  

^a EÖTVÖS LORÁND UNIVERSITY   (Hungary)

^b INSTITUTE OF EXPERIMENTAL MEDICINE   (Hungary)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33745702514     PISSN: 00472468     EISSN: 14208997     Source Type: Journal    
DOI: 10.1007/s00022-003-1702-2     Document Type: Article

References (80)

1. Abbott, H. L. and Liu, A., Property of B(s) and projective planes, Ars Combin. 20 (1985), 217-220.
2. Ball, S., Partial unitals and related structures in Desarguesian planes, Des. Codes Cryptogr. 15 (1998), 231-236.
3. Ball, S., The number of directions by a function over a finite field, manuscript, 2002.
4. Barat, J. and Innamorati, S., Largest minimal blocking sets in PG(2, 8), Journal of Comb. Designs 11 (3) (2003), 162-169.
5. Barat, J., Del Fra, A., Innamorati, S. and Storme, L., Minimal blocking sets in PG(2, 8) and maximal partial spreads in PG(3, 8), DCC, to appear.
6. Berardi, L. and Eugeni, F., Blocking sets e teoria dei giochi: origini e problematiche, Atti Sem. Mat. Fis. Univ. Modena 34 (1988), 165-196.
7. Berardi, L. and Eugeni, F., Blocking sets in the projective plane of order four, in: Combinatorics'86 (eds.: A. Barlotti et al.), Annals of Discrete Mathematics, 37 (1988), 43-50.
8. Berardi, L. and Innamorati, S., Irreducible blocking sets in the projective plane of order five, Atti Sem. Mat. Fis. Univ. Modena 38 (1990), 293-311.
9. Beres, T. and Illes, T., Computational investigation of the covering number of finite projective planes with small order (in Hungarian), Alk. Mat. Lapok 17 (1993/97), 397-111.
10. Bichara, A. and Korchmaros, G., n -sets in a projective plane which determine exactly n + n lines, J. Geom. 15 (1980), 175-181.
11. Bierbrauer, J., On minimal blocking sets, Arch. Math. 35 (1980), 394-400.
12. Blokhuis, A., On the size of a blocking set in PG(2, p), Combinatorica 14 (1994), 273-276.
13. Blokhuis, A., Blocking sets in Desarguesian planes, in: Paul Erdds is Eighty, Volume 2, (D. Miklos, V.T. Sos and T. Szonyi, eds.), Bolyai Soc. Math. Studies, 2, Bolyai Society, Budapest, 1996, 133-155.
14. Blokhuis, A., Ball, S., Brouwer, A. E., Storme, L. and Szonyi, T., On the number of slopes of the graph of a function defined on a finite field, J. Comb. Theory Ser. A 86 (1999), 187-196.
15. Blokhuis, A. and Brouwer, A. E., Blocking sets in Desarguesian projective planes, Bull. London Math. Soc. 18 (1986), 132-134
16. Blokhuis, A., Brouwer, A. E. and Szonyi, T., The number of directions determined by a function f on a finite field, J. Comb. Theory, Ser. A. 70 (1995), 349-353.
17. Blokhuis, A. and Metsch, K., Large minimal blocking sets, strong representative systems and partial unitals, in: Finite Geometries, (F De Clerck et al. eds.), Cambridge Univ. Press, Cambridge, 1993, 37-52.
18. Blokhuis, A., Pellikaan, R. and Szónyi, T, Blocking sets of almost Redei type, J. Comb. Theory Ser. A 78 (1997), 141-150.
19. Blokhuis, A., Brouwer, A. E. and Wilbrink, H. A., Blocking sets in PG(2, p) for small p, and partial spreads in PG(3, 7), Adv. Geom., to appear.
20. Brouwer, A. E. and Schrijver, A., The blocking number of an affine space, J. Combinatorial Theory Ser. A 24(1978), 251-253.
21. Brouwer, A. E. and Wilbrink, H. A., Blocking sets in translation planes, J. Geom. 19 (1982), 200.
22. Brown, J. M. N., Partitioning the complement of a simplex in PG(d, q), into copies of PG(d,q), J. Geom. 33 (1988), 11-16.
23. Bruen, A. A., Baer subplanes and blocking sets, Bull. Amer. Math. Soc. 76 (1970), 342-344.
24. Bruen, A. A., Blocking sets infinite projective planes, SLAM J. Appl. Math. 21 (1971), 380-392.
25. Bruen, A. A. and Thas, J. A., Blocking sets, Geom. Dedicata 6 (1977), 193-203.
26. Cameron, P. J., Four lectures on projective geometry, in: Finite Geometry (eds.: C.A. Baker, L.M. Batten), Lecture Notes in Pure Appl. Math. 103, M. Dekker, New York, 1985, 27-65.
27. Cornelissen, J. A., Ph. D Thesis, DIKU.
28. Cossidente, A., Gacs, A., Mengyan, Cs., Siciliano, A., Szonyi, T. and Weiner, Zs., Large minimal blocking sets in Galois planes of non-prime order, in preparation.
29. Drake, D. A., A bound for blocking sets infinite projective planes, in: Contemporary Math. Ill "Finite Geometries and Combinatorial Designs" (eds.: E. S. Kramer and S. S. Magliveras), Amer Math. Soc. 1991, 93-97.
30. Erdo's, P., Silverman, R. and Stein, A., Intersection properties of families of sets of nearly the same size, Ars Combin. 15 (1983), 247-259.
31. Füredi, Z., Matchings and covers in hypergraphs, Graphs and Combin. 4 (1988), 115-206.
32. Gács, A., On the number of directions determined by a point set in AG(2,p), Discrete Math. 208/9 (1999), 299-309.
33. Gács, A., A remark on blocking sets of almost Rédei type, J. Geom. 60 (1997), 65-73.
34. Gács, A., On a generalization of Rédei's theorem, Combinatorica, to appear.
35. Gács, A., Sziklai, P. and Szönyi, T., Two remarks on blocking sets and nuclei in planes of prime order, Designs, Codes, Cryptography 10 (1997), 29-39.
36. Gács, A. and Weiner, Zs., On (q + t,t)-arcs of type (0, 2,t), Des. Codes Cryptogr. 29 (2003), 131-139.
37. Graham, S. W. and Ringrose, C. J., Lower bounds for the least quadratic nonresidues, in: Analytic number theory, (Allerton Park, IL, 1989), Progr. in Math. Birkhuuser, 85 (1990), 269-309.
38. Hadnagy, É. and Szonyi, T., On the embedding of large (k, n)-arcs and partial unitals, Ars Combinatoria, 65 (2002), 299-308.
39. Hirschfeld, J. W P., Projective Geometries Over Finite Fields, Clarendon Press, Oxford, 1979, 2nd edition, 1998.
40. Hirschfeld, J. W. P. and Storme, L., The packing problem in statistics, coding theory and finite projective spaces, in: Finite Geometries (A. Blokhuis, J. W. P. Hirschfeld, D. Jungnickel, J. A. Thas, eds.) Developments of Mathematics, Kluwer, 2001, 201-246.
41. Hirschfeld, J. W. P. and Szonyi, T., Constructions of large arcs and blocking sets infinite planes, European J. Comb. 12 (1991), 499-511.
42. Illés, T. and Pisinger, D., Upper Bounds on the Covering Number of Galois-Planes with Small Order, J. Heuristics 7 (2000), 59-76.
43. Illés, T., Szonyi, T. and Wettl, F., Blocking sets and maximal strong representative systems infinite projective planes, Mitt. Math. Sem. Giessen 201 (1991), 97-107.
44. Innamorati, S., The non-existence of certain large minimal blocking sets, Mitt. Math. Sem. Giessen 235 (1998), 1-23.
45. Innamorati, S. and Maturo, A., On irreducible blocking sets in projective planes, Ratio Math. 2 (1991), 151-155.
46. Innamorati, S., Maturo, A., The spectrum of minimal blocking sets, Discrete Math. 208/209 (1999), 339-347.
47. Innamorati, S. and Zuanni, F., Minimal blocking configurations, J. Geom. 55 (1996), 86-98.
48. Kitto, C., A bound for blocking sets of maximal type in finite projective planes, Arch. Math. 52 (1989), 203-208.
49. Korchmáros, G. and Mazzocca, F., Nuclei of point sets of size q + 1 contained in the union of two lines in PG(2, q), Combinatorica 14 (1994), 63-69.
50. Lovasz, L. and Schrijver, A., Remarks on a theorem of Redei, Studia Sci. Math. Hungar. 16 (1981), 449-454.
51. Lunardon, G., Normal spreads, Geom. Dedicata 75 (1999), 245-261.
52. 1 1. J. Comb. Theory Ser. A 90 (2000), 148-158.
53. Lunardon, G. and Polverino, O., Blocking sets and derivable partial'spreads, J. Algebraic Comb. 14(2001), 49-56.
54. Lunardon, G. and Polverino, O., Linear blocking sets: a survey, in: Finite fields and applications (ed.: D. Jungnickel), Springer, 2001, 356-362.
55. Pambianco, F. and Storme, L., Minimal blocking sets in PG(2,9) and maximal partial spreads in PG(3,9), preprint.
56. di Paola, J., On minimum blocking coalitions in small projective plane games, SLAM J. Appl. Math. 17 (1969), 378-392.
57. Pelikán, J., Properties of incomplete balanced block designs, in: Combinatorial Theory and its Applications, (Balatonfured, Hungary, 1969), Colloq. Math. Soc. J. Bolyai 4 North Holland, Amsterdam, 1970, 869-889.
58. Polito, P. and Polverino, O., On small blocking sets, Combinatorica 18 (1998), 133-137.
59. 4), Australas. J. Combin. 26 (2002), 41-48.
60. Polverino, O., Small minimal blocking sets and complete k-arcs in PG(2, p), Discrete Math. 208/9 (1999), 469-476.
61. 3), Des. Codes Cryptogr. 20 (2000), 319-324.
62. Polverino, O., Blocking set nei piani proiettivi, Ph. D. Thesis, Univ. Naples, 1998.
63. Polverino, O. and Storme, L., Small minimal blocking sets in PG(2, q), Eur. J. Comb. 23 (2002), 83-92.
64. Polverino, O., Szonyi, T and Weiner, Zs., Blocking sets in Galois planes of square order, Acta Sci. Math. (Szeged) 65 (1999), 737-748.
65. Rédei, L., Lückenhafte Polynome über endlichen Körpern, Akadémiai Kiadó and Birkhäuser Verlag, Budapest and Basel, 1970.
66. Sherman, B. J., Minimal blocking sets in finite planes, J. of Geometry 43 (1992), 178-187.
67. Scherk, P., On the intersection number of two plane curves, J. of Geometry 10 (1977), 57-68.
68. Stöhr, K.-O. and Voloch, J. F., Weierstrass points and curves over finite fields, Proc. London Math. Soc. (3) 52 (1986), 1-19.
69. Sziklai, P. and Szonyi, T., Blocking sets and algebraic curves, Rend. Circ. Mat. Palermo 51 (1998), 71-86.
70. Szonyi, T., Combinatorial problems for abelian groups arising from geometry, Per. Polytechnica 19 (1991), 91-100.
71. Szonyi, T., Blocking sets infinite planes and spaces, Ratio Math. 5 (1992), 93-106.
72. Szonyi, T., Note on the existence of large minimal blocking sets in Galois planes, Combinatorica 12 (1992), 227-235.
73. Szonyi, T., Blocking sets in Desarguesian affine and projective planes, Finite Fields Appl. 3 (1997), 187-202.
74. Szonyi, T., Some applications of algebraic curves infinite geometry and combinatorics, in: Surveys in Combinatorics (R. A. Bailey, ed.), Cambridge Univ. Press, Cambridge, 1997, 198-236.
75. Szonyi, T., Around Redei's theorem, Discrete Math. 208/9 (1999), 557-575.
76. Szonyi, T. and Weiner, Zs., Large minimal blocking sets, manuscript, 2002.
77. Thas, J. A., Projective geometry over a finite field, Chapter in: Handbook of Incidence Geometry (F Buekenhout, ed.), North-Holland, Amsterdam, 1995.
78. Ughi, E., On (k., n)-fold blocking sets which can be obtained as a union of conies, Geom. Dedicata 26 (1988), 241-246.
79. Vedder, K., Affine subplanes of projective planes, in: Finite Geometries and designs, (Isle of Thorns, 1980), vol. 49 of London Math. Soc. Lecture Notes Series, Cambridge Univ. Press, 1981, 359-364.
80. Weiner, Zs., Blocking sets (in Hungarian), Diploma Thesis, Eotvos University, Budapest, 1998.