Alcuni problemi di ricerca nelle geometrie di Galois

Francesco Mazzocca

Bollettino dell'Unione Matematica Italiana (2001)

  • Volume: 4-B, Issue: 1, page 45-69
  • ISSN: 0392-4041

Abstract

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We present a survey on classical problems of Galois geometries. More precisely we discuss some problems and results about ovals, hyperovals, caps, maximal arcs and blocking sets in projective planes and spaces over Galois fields.

How to cite

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Mazzocca, Francesco. "Alcuni problemi di ricerca nelle geometrie di Galois." Bollettino dell'Unione Matematica Italiana 4-B.1 (2001): 45-69. <http://eudml.org/doc/195970>.

@article{Mazzocca2001,
author = {Mazzocca, Francesco},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {Galois geometries; ovals; ovoids; blocking sets; survey; research problems; maximal arcs; -caps},
language = {ita},
month = {2},
number = {1},
pages = {45-69},
publisher = {Unione Matematica Italiana},
title = {Alcuni problemi di ricerca nelle geometrie di Galois},
url = {http://eudml.org/doc/195970},
volume = {4-B},
year = {2001},
}

TY - JOUR
AU - Mazzocca, Francesco
TI - Alcuni problemi di ricerca nelle geometrie di Galois
JO - Bollettino dell'Unione Matematica Italiana
DA - 2001/2//
PB - Unione Matematica Italiana
VL - 4-B
IS - 1
SP - 45
EP - 69
LA - ita
KW - Galois geometries; ovals; ovoids; blocking sets; survey; research problems; maximal arcs; -caps
UR - http://eudml.org/doc/195970
ER -

References

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  1. ARTIN, E., Geometric Algebra, Interscience, 1957. Zbl0077.02101MR82463
  2. ASSMUS, E. F.- KEY, E. F., Designs and their Codes, Cambridge Tracts in Mathematics, 103, Cambridge University Press, 1992. Zbl0762.05001MR1192126
  3. BALL, S.- BLOKHUIS, A.- MAZZOCCA, F., Maximal arcs in Desarguesian planes of odd order do not exist, Combinatorics, 17 (1997), 31-41. Zbl0880.51003MR1466573
  4. BARLOTTI, A., Un'estensione del teorema di Segre-Kustaanheimo, Boll. U.M.I., 10, Serie III (1955), 498-506. Zbl0066.38901MR75606
  5. BERARDI, L., Algebra e teoria dei codici correttori, Collana di matematica e statistica, Franco Angeli, 1994. 
  6. BERARDI, L.- EUGENI, F., Blocking sets e teoria dei giochi: origini e problematiche, Atti Sem. Fis., Univ. Modena, 34 (1988), 165-196. Zbl0656.51011MR956788
  7. BEUTELSPACHER, A., Einfurung in die endliche geometrie, vol. I, II, Wiessensnaftsverlag, 1983. Zbl0502.51001MR735433
  8. BEUTELSPACHER, A.- ROSENBAUM, U., Projective Geometry, Cambridge University Press, 1998. Zbl0893.51001MR1629468
  9. BIGGS, N. L.- WHITE, A. T., Permutation Groups and Combinatorial Structures, London Mathematical Society, Lecture Note Series, 33, Cambridge University Press, 1979. Zbl0415.05002MR540889
  10. BLOKHUIS, A., Blocking sets in Desarguesian planes, Combinatorics, Paul Erdos is Eighty (vol. 2), Bolyai Society Mathematical Studies, Keszthely (Ungheria), 1-20, 1993. Zbl0849.51005MR1395857
  11. BLOKHUIS, A.- MAZZOCCA, F., On maximal sets of nuclei in P G 2 , q and quasi-odd sets in A G 2 , q , Advances in finite geometries and designs, Oxford University Press, 35-46, 1991. Zbl0737.51005MR1138731
  12. BLOKHUIS, A.- MAZZOCCA, F., Special point sets in P G n , q and the structure of sets with the maximal number of nuclei, J. Geom., 41 (1991), 33-41. Zbl0737.51009MR1116900
  13. BLOKHUIS, A.- MAZZOCCA, F., Lifts of nuclei in finite projective spaces, L.M.S. Lecture Note Series, 191 (1993), 31-36. Zbl0801.51002MR1256261
  14. BLOKHUIS, A.- WILBRINK, H. A., A characterization of exterior lines of certain sets of points in P G 2 , q , Geom. Dedicata, 23 (1987), 253-254. Zbl0621.51013MR892405
  15. BOSE, R. C., Mathematical Theory of the Symmetrical Factorial Design, Sankhya, 8 (1947), 107-166. Zbl0038.09601MR26781
  16. BOSE, R. C., On some connections between the design of experiments and information theory, Bull. Inst. Internat. Statist., 38 (1964), 257-271. Zbl0102.14503MR148190
  17. BOSE, R. C.- SRIVASTAVA, J. N., On a bound useful in the theory of factorial designs and error correcting codes, Ann. Math. Statist., 35 (1964), 408-414. Zbl0124.11503MR159399
  18. BROUWER, A. E.- SCHRIJVER, A., The blocking number of an affine space, J. of Combin. Theory (A), 24 (1978), 251-253. Zbl0373.05020MR480094
  19. BROWN, M. R., Ovoids of P G 3 , q , q even, with a conic section, J. London Math. Soc., to appear. MR1783645
  20. BRUEN, A. A., Baer subplanes and blocking sets, Bull. Amer. Math. Soc., 76 (1970), 342-344. Zbl0207.02601MR251629
  21. BRUEN, A. A., Blocking sets in finite projective planes, SIAM J. of Appl. Math., 21 (1971), 380-392. Zbl0252.05014MR303406
  22. BRUEN, A. A., Nuclei of sets of q + 1 points in P G 2 , q and blocking sets of Rédei type, J. of Combin. Theory (A), 55 (1990), 130-132. Zbl0706.51011MR1070019
  23. BRUEN, A. A.- DE RESMINI, M. J., Blocking sets in affine planes, Annals of Discrete Math., 18 (1983), 169-176. Zbl0508.05021MR695804
  24. BRUEN, A. A.- THAS, J. A., Flocks, chains and configurations in finite geometries, Atti Accad. Naz. Lincei, Rend. Cl. Fis. e Mat., 59 (1975), 744-748. Zbl0356.50019MR487763
  25. BRUEN, A. A.- THAS, J. A., Blocking sets, Geom. Ded., 6 (1977), 193-203. Zbl0367.05009MR461271
  26. CAMERON, P. J., Four lectures on projective geometry, in Finite Geometries, Lecture Notes in Pure and Applied Mathematics, 103 (1985), 27-65. Zbl0591.51012MR826794
  27. CAMERON, P. J.- VAN LINT, J. H., Designs, Graphs, Codes and their Links, London Mathematical Society, Student Texts, 22, 1991. Zbl0743.05004MR1148891
  28. CAMERON, P. J.- MAZZOCCA, F., Bijections which preserve blocking sets, Geom. Dedicata, 21 (1986), 219-229. Zbl0619.05001MR861199
  29. CERASOLI, M.- EUGENI, F.- PROTASI, M., Elementi di matematica discreta, Zanichelli, 1988. Zbl0746.00001MR1093460
  30. CHEROWITZO, W., Hyperovals in Desarguesian planes of even order, Ann. Discrete Math., 37 (1988), 87-94. Zbl0651.51009MR931308
  31. CHEROWITZO, W.- PENTTILA, T.- PINNERI, I., Flocks and ovals, Geom. Dedicata, 60 (1996), 17-37. Zbl0855.51008MR1376478
  32. CHEROWITZO, W.- O'KEEFE, C. M.- PENTTILA, T., A unified construction of finite geometries in characteristic two, Preprint. 
  33. COSSU, A., Su alcune proprietà dei k , n -archi di un piano proiettivo sopra un corpo finito, Rend. Mat. e Appl., 20 (1961), 271-277. Zbl0103.38106MR139995
  34. COXETER, H. S. M., Projective geometry, Springer, Second Edition, 1987. Zbl0629.51001MR346652
  35. DEMBOWSKI, P., Finite geometries, Springer-Verlag, 1968. Zbl0159.50001MR233275
  36. DENNISTON, R. H. F., Some maximal arcs in finite projective planes, J. Combinatorial Theory, 6 (1969), 317-319. Zbl0167.49106MR239991
  37. FELLEGARA, G., Gli ovaloidi di uno spazio tridimensionale di Galois di ordine 8, Atti Acc. Naz. Lincei, Rend. Cl. Sci. Mat., Fis., Nat., 32 (1962), 170-176. Zbl0109.39403MR155224
  38. FISHER, R. A., The theory of confounding in factorial experiments in relation to the theory of groups, Ann. Eugen. London, 11 (1942), 341-352. Zbl0060.31502MR7382
  39. FISHER, R. A., A system confounding for factors with more than two alternatives giving completely orthogonal cubes and higher powers, Ann. Eugen. London, 12 (1945), 283-290. Zbl0060.31503MR13113
  40. GLYNN, D. G., Two new sequences of ovals in finite Desarguesian Planes of even order, Combinatorial Mathematics X (Lecture Note in Math.1036, Springer), edited by R. Casse, 217-229, 1983. Zbl0531.51010MR731584
  41. HAMMING, R. W., Error detecting and error correcting codes, Bell Syst. Tech. J., 29 (1950), 147-160. MR35935
  42. HILL, R., On the larget size of cap in S 5 , 3 , Atti Accad. Naz. Lincei, Rend. Cl. Sc. Mat., Fis., Nat., 54 (1973), 378-384. Zbl0342.05017MR363951
  43. HIRSCHFELD, J. W. P., Projecttive geometries over finite fields, Oxford University Press, Oxford, 1979. Zbl0418.51002MR554919
  44. HIRSCHFELD, J. W. P., Finite projective spaces of three dimensions, Clarendon Press, Oxford, 1985. Zbl0574.51001MR840877
  45. HIRSCHFELD, J. W. P.- THAS, J. A., General Galois Geometries, Oxford Mathematical Monographs, Oxford University Press, 1991. Zbl0789.51001MR1363259
  46. HUGHES, D. R., On t -designs and groups, Amer. J. Math., 87 (1965), 761-768. Zbl0134.03004MR188288
  47. HUGHES, D. R.- PIPER, F. C., Projective planes, Springer Verlag, 1973. Zbl0267.50018MR333959
  48. HUGHES, D. R.- PIPER, F. C., Design Theory, Cambridge University Press, 1988. Zbl0659.05008MR938742
  49. JAMISON, R., Covering finite fields with cosets of subspaces, J. of Combin. Theory (A), 22 (1977), 253-266. Zbl0354.12019MR439664
  50. JÄRNEFELT, G.- KUSTAANHEIMO, P., An observation on finite geometries, Comptes Rendus XI Congr. Math. Scand., Trodheim, 166-182, 1949. Zbl0048.37103MR54979
  51. KORCHMAROS, G.- MAZZOCCA, F., On q + t -arcs of type 0 , 2 , t in a Desarguesian plane of order q i , Mathematical Proceedings of the Cambridge Philosophical Society, 108 (1990), 445-449. MR1068447
  52. LAM, C. W. H.- THIEL, L.- SWIERCZ, S.- MCKAY, J., The non-existence of ovals in a projective plane of order 10, Discrete Math., 45 (1983), 319-321. Zbl0574.05008MR704249
  53. LAM, C. W. H.- THIEL, L.- SWIERCZ, S., The non-existence of finite projective planes of order 10, Canad. J. Math., 41 (1989), 1117-1123. Zbl0691.51003MR1018454
  54. LUNELLI, L.- SCE, M., k -Archi completi nei piani proiettivi desarguesiani di rango 8 e 16. Centro di Calcoli numerici, Politecnico di Milano, 1958. MR157276
  55. MAZZOCCA, F., Blocking sets with respect to special families of lines and nuclei of θ n -sets in finite n -dimensional projective and affine spaces, Mitt. Math. Sem. Giessen, 201 (1991), 109-117. Zbl0741.51014MR1126310
  56. VON NEUMANN, J.- MORGENSTERN, O., Theory of games and economic behavior, Princeton, 1947. Zbl0053.09303MR21298
  57. PANELLA, G., Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito, Boll. U.M.I., 10, Serie III (1955), 507-513. Zbl0066.38902MR75607
  58. DI PAOLA, J., On a restricted class of block design games, Canadian J. Math., 18 (1966), 225-236. Zbl0228.05013MR194207
  59. DI PAOLA, J., On minimum blocking coalitions in small projective games, SIAM J. of Appl. Math., 17 (1969), 378-392. Zbl0191.49601MR247879
  60. PAYNE, S. E., A new infinite family of generalized quadrangles, Congr. Numer., 49 (1985), 115-128. Zbl0632.51011MR830735
  61. PELLEGRINO, G., Sul massimo ordine delle calotte in S 4 , 3 , Matematiche (Catania), 25 (1970), 1-9. MR363952
  62. PENTTILA, T.- ROYLE, G. F., Sets of type m , n in the affine and projective planes of order nine, Designs, Codes and Cryptography, 6 (1995), 229-245. Zbl0841.05012MR1351846
  63. QVIST, B., Some remarks concerning curves of the second degree in a finite plane, Ann. Acad. Sci. Fenn., Ser. A, 134, 1952. Zbl0049.10801MR54977
  64. RÉDEI, L., Lacunary polynomials over finite fields, North Holland, Amsterdam, 1973. Zbl0255.12009MR352060
  65. RICHARDSON, M., On finite projective games, Proc. Amer. Math. Soc., 7 (1956), 458-465. Zbl0073.35403MR79543
  66. SCAFATI, M.- TALLINI, G., Geometria di Galois e teoria dei codici, CISU, Roma, 1995. 
  67. SEGRE, B., Lezioni di geometria moderna, vol. I, Zanichelli, Bologna, 1948. Zbl0030.41005MR30204
  68. SEGRE, B., Sulle ovali dei piani lineari finiti, Atti Acc. Naz. Lincei, Rend. Cl. Sci. Mat., Fis., Nat., 17 (1954), 141-142. Zbl0057.36301MR71035
  69. SEGRE, B., Ovals in finite projective planes, Canad. J. Math., 7 (1955), 414-416. Zbl0065.13402MR71034
  70. SEGRE, B., Sui k -archi nei piani finiti di caratteristica due, Rev. Math. Pures Appl., 2 (1957), 289-300. Zbl0087.35003MR98347
  71. SEGRE, B., On complete caps and ovaloids in three dimensional Galois spaces of characteristic two, Acta Arith., 5 (1959), 315-332. Zbl0094.15902MR114153
  72. SEGRE, B., Le geometrie di Galois, Ann. Mat. Pura e Appl., 48 (1959), 1-97. Zbl0093.33604MR116259
  73. SEGRE, B., Lectures on modern geometry, Ed. Cremonese, Roma, 1961. Zbl0095.14802MR131192
  74. SEGRE, B., Ovali e curve σ nei piani di Galois di caratteristica due, Atti Acc. Naz. Lincei, Rend. Cl. Sci. Mat., Fis., Nat., 33 (1962), 785-790. Zbl0107.38102MR149361
  75. SEGRE, B.- BARTOCCI, U., Ovali ed altre curve nei piani di Galois di caratteristica due, Acta Arith., 18 (1971), 423-449. Zbl0219.50016MR295201
  76. SEGRE, B.- KORCHMAROS, G., Una proprietà degli insiemi di punti di un piano di Galois caratterizzante quelli formati dai punti delle singole rette esterne ad una conica, Atti Accad. Naz. Lincei, Rend. Cl. Fis. e Mat., 62 (1977), 1-7. Zbl0383.51006MR513866
  77. TALLINI, G., Sulle k -calotte di uno spazio lineare finito, Ann. Mat., 42 (1956), 119-164. Zbl0074.15304MR85531
  78. TALLINI, G., Partial line spaces and algebraic varieties, Symposia Mathematica, XXVIII (1986), 203-217. Zbl0616.51019MR873781
  79. TALLINI, G., Teoria dei k -insiemi in uno spazio di Galois. Teoria dei codici correttori, Quaderno n. 64, Sem. Geom. Comb. Dip. Mat. Univ. Roma La Sapienza, 1985. 
  80. THAS, J. A., Connection between the Grassmannian G k - 1 ; n and the set of the k -arcs of the Galois space S n , q , Rend. Mat., 2 (1969), 121-134. Zbl0232.50013
  81. THAS, J. A., Some results concerning q + 1 n - 1 ; n -arcs and q + 1 n - 1 + 1 ; n -arcs in finite projective planes of order q , J. Comb Theory, Ser. A, 19 (1975), 228-232. Zbl0311.50016
  82. TITS, J., Les groupes simples de Suzuki et de Ree, Seminaire Bourbaki, 210, 1960. Zbl0267.20041MR1611778
  83. TITS, J., Ovoides et groupes du Suzuki, Arch. Math., 13 (1962), 187-198. Zbl0109.39402MR140572
  84. TITS, J., Un proprieté caracteristique des ovoïdes associés aux groupes de Suzuki, Arch. Math., 17 (1966), 136-153. Zbl0136.15701MR203564
  85. TONCHEV, V. D., Combinatorial Configurations, Longman Scientific & Technical, 1988. Zbl0545.05002

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