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On the spectrum of the valuesk for which a completek- cap in PG(n, q) exists

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Abstract

the aim of this paper is to collect all results on the spectrum of values k that occur as the cardinality of a complete k- cap in a finite projective space. 1

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References

  1. Abatangelo, V.:Complete arcs in PG(2,q), q even. Annals Discrete Math.18 (1983), 13–15.

    Google Scholar 

  2. Abatangelo, V.:A class of complete ((q + 8)/3)-arcsof PG(2,q)with q= 2h and h (≥6)even. Ars Combinatoria16 (1983), 103–111.

    Google Scholar 

  3. Abatangelo, V.,Korchmáros, G. andLarato, B.:Classification of maximal caps in PG(3, 5). To appear in J. Geom.

  4. Ball, S.:On sets of points in finite planes. Ph. D. Thesis, Univ. of Sussex, England, 1994.

    Google Scholar 

  5. Barlotti, A.:Un'estensione del teorema di Segre-Kustaanheimo. Boll. Un. Mat. Ital.10 (1955), 96–98.

    Google Scholar 

  6. Barlotti, A.:Some topics in finite geometrical structures. Inst. of Statistics mimeo series no. 439, Univ. of North Carolina (1965).

  7. Blokhuis, A.:Note on the size of a blocking set in PG(2,q). Combinatorica14 (1994), 111–114.

    Google Scholar 

  8. Boros, E. andSzönyi, T.:On the sharpness of a theorem of B. Segre. Combinatorica6 (1986), 261–268.

    Google Scholar 

  9. Bose, R.C.:Mathematical theory of the symmetrical factorial design. Sankhya8 (1947), 107–166.

    Google Scholar 

  10. Davidov, A.A. andTombak, L.M.:Quasiperfect linear binary codes with distance 4 and complete caps in projective geometry. Problems of Inform. Transmission25 (1990), 265–275.

    Google Scholar 

  11. Faina, G.:Complete caps having less than (q2 + 1)/2points in common with an elliptic quadric of PG(3,q). Rend. Mat. Appl.8 (1988), 277–281.

    Google Scholar 

  12. Faina, G.:Complete k-caps in PG(3,q)with k < (q2 + q + 4)/2. Ars Combinatoria33 (1992), 311–317.

    Google Scholar 

  13. Faina, G.:The maximum size of a (Ω,P, Q)-cap in PG(3, 5). Combinatorics '88 (Proc. International Conference on Incidence Geometries and Combinatorial Structures) Mediterranean Press 1991 (Vol. I), 373–380.

  14. Faina, G.:Il packing problem nette geometrie di Galois. Atti XV Congresso Unione Matematica Italiana, Padova, 11–16 Settembre 1995.

  15. Faina, G.,Marcugini, S.,Milani, A. andPambianco, F.:The sizes of the complete arcs in PG(2,q)for q ≤ 23. Designs, Codes and Cryptography. Submitted.

  16. Faina, G.,Marcugini, S.,Milani, A. andPambianco, F.:The sizes of the complete k-caps in PG(n, q),for small q and 3≤n≤5. Ars Combinatoria. Submitted.

  17. Faina, G. andPambianco, F.:A class of complete k-caps in PG(3,q)for q an odd prime. To appear in J. Geom.

  18. Faina, G. andPambianco, F.:Small complete k-caps of PG(r, q),r ≥ 3. To appear in Annals Discrete Math.

  19. Faina, G. andPambianco, F.:On 10-arcs in projective planes, Preprint.

  20. Fellegara, G.:Gli ovaloidi di uno spazio tridimensionale di Galois di or dine 8. Atti Accad. Naz. Lincei Rend.32 (1962), 170–176.

    Google Scholar 

  21. Fisher, J.C., Hirschfeld J.W.P. andThas J.A.:Complete arcs in planes of square order. Annals Discrete Math.30 (1986), 243–250.

    Google Scholar 

  22. Gabidulin, E.M., Davidov, A.A. andTombak, L.M.:Codes with covering radius 2 and other new covering codes. IEEE Trans. Inform. TheoryIT-37 (1991), 219–224.

    Google Scholar 

  23. Giulietti, M. andUghi, E.:Un arco completo di cardinalità relativamente piccola in PG(2,q),q= p2,p ≡ 3 (mod 4). Rapporto Tecnico n. 14/95, Università degli Studi di Perugia (1995).

  24. Gordon, C.E.:Orbits of arcs in PG(N, K)under projectivities. Geom. Dedicata42 (1992), 187–203.

    Google Scholar 

  25. Gronchi, P.:I blocking sets ed il packing problem. Bull. Un. Mat. Ital.7-A (1993), 227–236.

    Google Scholar 

  26. Hill, R.:On the largest size of a cap in S5,3. Atti Accad. Naz. Lincei Rend.54 (1973), 378–384.

    Google Scholar 

  27. Hill, R.:Caps and codes. Discrete Math.22 (1978), 111–137.

    Google Scholar 

  28. Hill, R.:On Pellegrino's 20-capsin 4,3. Annals Discrete Math.18 (1983), 443–448.

    Google Scholar 

  29. Hirschfeld, J.W.P.:Projective Geometries over Finite Fields. Clarendon Press, Oxford (1979).

    Google Scholar 

  30. Hirschfeld, J.W.P.:Maximum sets in finite projective spaces. Surveys in Combinatorics (E.K. Lloyd ed.), London Math. Soc. L.N. Series82, Cambridge Univ. Press (1983), 55–76.

  31. Hirschfeld, J.W.P.:Finite projective spaces of three dimensions. Clarendon Press, Oxford (1985).

    Google Scholar 

  32. Hirschfeld, J.W.P. andStorme, L.:The packing problem in statistics, coding theory and finite projective spaces. Preprint.

  33. Hirschfeld, J.W.P. andThas, J.A.:General Galois Geometries. Oxford Univ. Press, Oxford (1991).

    Google Scholar 

  34. Hirschfeld, J.W.P. andThas, J.A.:Linear independence in finite spaces. Geometriae Dedicata23 (1987), 15–31.

    Google Scholar 

  35. Kaneta, H.:Private communication (1995).

  36. Kestenband, B.C.:Unitals intersections in finite projective planes. Geom. Dedicata.11 (1981), 107–117.

    Google Scholar 

  37. Kestenband, B.C.:A family of complete k-arcs in PG(2,q). Colloq. Math.62 (1989), 59–67.

    Google Scholar 

  38. Korchmàros, G.:Osservazioni sui risultati di B. Segre relativi ai k-archi completi contenenti k-1punti di una conica. Atti Accad. Naz. Lincei Rend.56 (1974), 541–549.

    Google Scholar 

  39. Korchmàros, G.:New examples of k-arcs in PG(2,q). European J. Comb.4 (1983), 329–334.

    Google Scholar 

  40. Lisonek, P.:Computer-assisted studies in algebraic combinatorics. Ph.D. Thesis, Research Institute for Symbolic Computation, J. Kepler Univ. Linz, 1994. RISC-Linz Report Series No. 94–68.

    Google Scholar 

  41. Lombardo Radice, L.:Sul problema dei k-archi completi di S2,q. Boll. Un. Mat. Ital.11 (1956), 178–181.

    Google Scholar 

  42. Lunelli, L. andSce, M.:Considerazioni aritmetiche e risultati sperimentali sui {k;nq− archi. Ist. Lombardo Accad. Sci. Rend.A 98 (1964), 3–52.

    Google Scholar 

  43. Marcugini, S.,Milani, A. andPambianco, F.:A computer search for complete arcs in PG(2,q),q ≤ 128. Rapporto Tecnico n. 18/95, Università degli Studi di Perugia (1995).

  44. Marcugini, S.,Milani, A. andPambianco, F.:A computer search for complete caps in PG(d,q). Rapporto Tecnico n. 19/95, Università degli Studi di Perugia (1995).

  45. O'keefe, C.M. andPenttila, T.:Ovoids of PG(3, 16)are elliptic quadrics. J. Geom.38 (1990), 95–106.

    Google Scholar 

  46. O'keefe, C.M. andPenttila, T.:A new hyperoval in PG(2, 32). J. Geom.44 (1992), 117–139.

    Google Scholar 

  47. O'keefe, C.M. andPenttila, T.:Ovoids of PG(3, 16)are elliptic quadrics. II. J. Geom.44 (1992), 140–159.

    Google Scholar 

  48. O'keefe, C.M., Penttila, T. andRoyle, G.F.:Classification of ovoids in PG(3, 32). J. Geom.50 (1994), 143–150.

    Google Scholar 

  49. Pambianco, F. andStorme, L.:Small complete caps in spaces of even characteristic. To appear in J. Comb. Theory, Ser. A.

  50. Panella, G.:Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito. Boll. Un. Mat. Ital.10 (1955), 507–513.

    Google Scholar 

  51. Pellegrino, G.:Sul massimo or dine delle calotte di S4,3. Le Matematiche (Catania)25 (1970), 1–9.

    Google Scholar 

  52. Pellegrino, G.:Un'osservazione sul problema dei k-archi completi di S2,q con q ≡ 1 (mod 4). Atti Accad. Naz. Lincei Rend.63 (1977), 33–34.

    Google Scholar 

  53. Pellegrino, G.:Archi completi di ordine (q + 3)/2nei piano di Galois S2,q con q ≡ 3 (mod 4). Rend. Mat.2 (1982), 59–66.

    Google Scholar 

  54. Pellegrino, G.:Sur les k-arcs completes des planes de Galois d'ordre impair. Annals Discrete Math.18 (1983), 667–694.

    Google Scholar 

  55. Pellegrino, G.:Sugli archi completi dei piani di Galois, di ordine dispari, contenenti (q + 3)/2punti di un ovale. Rend. Sem. Mat. Univ. Brescia.7 (1984), 495–523.

    Google Scholar 

  56. Pellegrino, G.:Archi completi contenenti (q + l)/2punti di una conica, nei piani di Galois di ordine dispari. Rend. Circolo Mat. Palermo42 (1993), 273–308.

    Google Scholar 

  57. Pellegrino, G.:Sulle calotte complete, non ovaloidi, dello spazio PG(3,q),q dispari. To appear in Annals Discrete Math.

  58. Penttila, T. andPraegher, C.E.:Ovoids and translation ovals. Preprint.

  59. Penttila, T. andRoyle, G.F.:Private communication (1995).

  60. Qvist, B.:Some remarks concerning curves of the second degree in a finite plane. Ann. Acad. Sci. Fenn. Ser. A134 (1952), 137–236.

    Google Scholar 

  61. Royle, G.:Private communication (1995).

  62. Segre, B.:Ovals in a finite projective plane. Canad. J. Math.7 (1955), 414–416.

    Google Scholar 

  63. Segre, B.:Curve razionali normali e k-archi negli spazi finiti. Ann. Mat. Pura Appl.39 (1955), 357–379.

    Google Scholar 

  64. Segre, B.:On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two. Acta Aritm.5 (1959), 315–332.

    Google Scholar 

  65. Segre, B.:Le geometrie di Galois. Ann. Mat Pura Appl.48 (1959), 1–97.

    Google Scholar 

  66. Segre, B.:Introduction to Galois Geometries. Mem Accad. Naz. Lincei8 (1967), 137–236.

    Google Scholar 

  67. Seiden, B.:A theorem infinite projective geometry and an application to Statistics. Proc. Amer. Math. Soc.1 (1950), 282–286.

    Google Scholar 

  68. Storme, L. andSzönyi, T.:Caps in PG(n, q), q even, n ≥ 3. Geometriae Dedicata45 (1993), 163–169.

    Google Scholar 

  69. Szönyi, T.:Arcs, caps, codes, and 3-independent subsets. Proc. International Conference “Giornate di Geometrie Combinatorie” (eds.: G. Faina and G. Tallini) Perugia 1992, Università degli Studi di Perugia (1993), 57–80.

  70. Szönyi, T.:Complete arcs in Galois planes: a survey. Quaderni del Sem. Geom. Comb. Univ. Roma94 (1989).

  71. Tallini, G.:Le geometrie di Galois e le loro applicazioni alla statistica e alla teoria delle informazioni. Rend. Mate Appl.19 (1960), 379–400.

    Google Scholar 

  72. Tallini, G.:Calotte complete di S4 contenenti due quadriche ellittiche quali sezioni iperpiane. Rend. Mat e Appl.23 (1964), 108–128.

    Google Scholar 

  73. Tallini, G.:Teoria dei k-insiemi in uno spazio di Galois. Teoria dei codici correttori. Seminario di Geom. Comb. Univ. Roma. no.64 (1985).

  74. Tallini Scafati, M.:Archi completi in S2,q con q pari. Atti Accad. Naz. Lincei Rend.37 (1964), 48–51.

    Google Scholar 

  75. Tits, J.:Les groupes simples de Suzuki et de Ree. Sèm. Bourbaki210 (1960), 1–18.

    Google Scholar 

  76. Tits, J.:Ovoïdes à translation. Rend. Mat. e Appl.21 (1962), 37–59.

    Google Scholar 

  77. Tits, J.:Ovoïdes et groupes de Suzuki. Arch. Math.13 (1962), 187–198.

    Google Scholar 

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  1. Dipartimento di Matematica, Università degli Studi di Perugia, Via Vanvitelli 1, 06100, Perugia, Italy

    Giorgio Faina & Fernanda Pambianco

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  1. Giorgio Faina
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Faina, G., Pambianco, F. On the spectrum of the valuesk for which a completek- cap in PG(n, q) exists. J Geom 62, 84–98 (1998). https://doi.org/10.1007/BF01237602

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