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Mazzocca, Francesco:
Alcuni problemi di ricerca nelle geometrie di Galois
Bollettino dell'Unione Matematica Italiana Serie 8 4-B (2001), fasc. n.1, p. 45-69, Unione Matematica Italiana (italian)
pdf (472 Kb), djvu (312 Kb). | MR1821397 | Zbl 1036.51008

Sunto

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.
Referenze Bibliografiche
[1] E. ARTIN, Geometric Algebra, Interscience, 1957. | MR 82463 | Zbl 0077.02101
[2] E. F. ASSMUS-E. F. KEY, Designs and their Codes, Cambridge Tracts in Mathematics, 103, Cambridge University Press, 1992. | MR 1192126 | Zbl 0762.05001
[3] S. BALL-A. BLOKHUIS-F. MAZZOCCA, Maximal arcs in Desarguesian planes of odd order do not exist, Combinatorics, 17 (1997), 31-41. | MR 1466573 | Zbl 0880.51003
[4] A. BARLOTTI, Un'estensione del teorema di Segre-Kustaanheimo, Boll. U.M.I., 10, Serie III (1955), 498-506. | fulltext bdim | MR 75606 | Zbl 0066.38901
[5] L. BERARDI, Algebra e teoria dei codici correttori, Collana di matematica e statistica, Franco Angeli, 1994.
[6] L. BERARDI-F. EUGENI, Blocking sets e teoria dei giochi: origini e problematiche, Atti Sem. Fis., Univ. Modena, 34 (1988), 165-196. | MR 956788 | Zbl 0656.51011
[7] A. BEUTELSPACHER, Einfurung in die endliche geometrie, vol. I, II, Wiessensnaftsverlag, 1983. | MR 735433 | Zbl 0502.51001
[8] A. BEUTELSPACHER-U. ROSENBAUM, Projective Geometry, Cambridge University Press, 1998. | MR 1629468 | Zbl 0893.51001
[9] N. L. BIGGS-A. T. WHITE, Permutation Groups and Combinatorial Structures, London Mathematical Society, Lecture Note Series, 33, Cambridge University Press, 1979. | MR 540889 | Zbl 0415.05002
[10] A. BLOKHUIS, Blocking sets in Desarguesian planes, Combinatorics, Paul Erdos is Eighty (vol. 2), Bolyai Society Mathematical Studies, Keszthely (Ungheria), 1-20, 1993. | MR 1395857 | Zbl 0849.51005
[11] A. BLOKHUIS-F. MAZZOCCA, On maximal sets of nuclei in $PG(2, q)$ and quasi-odd sets in $AG(2, q)$, Advances in finite geometries and designs, Oxford University Press, 35-46, 1991. | MR 1138731 | Zbl 0737.51005
[12] A. BLOKHUIS-F. MAZZOCCA, Special point sets in $PG(n, q)$ and the structure of sets with the maximal number of nuclei, J. Geom., 41 (1991), 33-41. | MR 1116900 | Zbl 0737.51009
[13] A. BLOKHUIS-F. MAZZOCCA, Lifts of nuclei in finite projective spaces, L.M.S. Lecture Note Series, 191 (1993), 31-36. | MR 1256261 | Zbl 0801.51002
[14] A. BLOKHUIS-H. A. WILBRINK, A characterization of exterior lines of certain sets of points in $PG(2, q)$, Geom. Dedicata, 23 (1987), 253-254. | MR 892405 | Zbl 0621.51013
[15] R. C. BOSE, Mathematical Theory of the Symmetrical Factorial Design, Sankhya, 8 (1947), 107-166. | MR 26781 | Zbl 0038.09601
[16] R. C. BOSE, On some connections between the design of experiments and information theory, Bull. Inst. Internat. Statist., 38 (1964), 257-271. | MR 148190 | Zbl 0102.14503
[17] R. C. BOSE-J. N. SRIVASTAVA, On a bound useful in the theory of factorial designs and error correcting codes, Ann. Math. Statist., 35 (1964), 408-414. | fulltext mini-dml | MR 159399 | Zbl 0124.11503
[18] A. E. BROUWER-A. SCHRIJVER, The blocking number of an affine space, J. of Combin. Theory (A), 24 (1978), 251-253. | MR 480094 | Zbl 0373.05020
[19] M. R. BROWN, Ovoids of $PG(3, q)$, $q$ even, with a conic section, J. London Math. Soc., to appear. | MR 1783645
[20] A. A. BRUEN, Baer subplanes and blocking sets, Bull. Amer. Math. Soc., 76 (1970), 342-344. | fulltext mini-dml | MR 251629 | Zbl 0207.02601
[21] A. A. BRUEN, Blocking sets in finite projective planes, SIAM J. of Appl. Math., 21 (1971), 380-392. | MR 303406 | Zbl 0252.05014
[22] A. A. BRUEN, Nuclei of sets of $q+1$ points in $PG(2, q)$ and blocking sets of Rédei type, J. of Combin. Theory (A), 55 (1990), 130-132. | MR 1070019 | Zbl 0706.51011
[23] A. A. BRUEN-M. J. DE RESMINI, Blocking sets in affine planes, Annals of Discrete Math., 18 (1983), 169-176. | MR 695804 | Zbl 0508.05021
[24] A. A. BRUEN-J. A. THAS, Flocks, chains and configurations in finite geometries, Atti Accad. Naz. Lincei, Rend. Cl. Fis. e Mat., 59 (1975), 744-748. | MR 487763 | Zbl 0356.50019
[25] A. A. BRUEN-J. A. THAS, Blocking sets, Geom. Ded., 6 (1977), 193-203. | MR 461271 | Zbl 0367.05009
[26] P. J. CAMERON, Four lectures on projective geometry, in Finite Geometries, Lecture Notes in Pure and Applied Mathematics, 103 (1985), 27-65. | MR 826794 | Zbl 0591.51012
[27] P. J. CAMERON-J. H. VAN LINT, Designs, Graphs, Codes and their Links, London Mathematical Society, Student Texts, 22, 1991. | MR 1148891 | Zbl 0743.05004
[28] P. J. CAMERON-F. MAZZOCCA, Bijections which preserve blocking sets, Geom. Dedicata, 21 (1986), 219-229. | MR 861199 | Zbl 0619.05001
[29] M. CERASOLI-F. EUGENI-M. PROTASI, Elementi di matematica discreta, Zanichelli, 1988. | MR 1093460 | Zbl 0746.00001
[30] W. CHEROWITZO, Hyperovals in Desarguesian planes of even order, Ann. Discrete Math., 37 (1988), 87-94. | MR 931308 | Zbl 0651.51009
[31] W. CHEROWITZO-T. PENTTILA-I. PINNERI, Flocks and ovals, Geom. Dedicata, 60 (1996), 17-37. | MR 1376478 | Zbl 0855.51008
[32] W. CHEROWITZO-C. M. O'KEEFE-T. PENTTILA, A unified construction of finite geometries in characteristic two, Preprint.
[33] A. COSSU, Su alcune proprietà dei $\{k, n\}$-archi di un piano proiettivo sopra un corpo finito, Rend. Mat. e Appl., 20 (1961), 271-277. | MR 139995 | Zbl 0103.38106
[34] H. S. M. COXETER, Projective geometry, Springer, Second Edition, 1987. | MR 346652 | Zbl 0629.51001
[35] P. DEMBOWSKI, Finite geometries, Springer-Verlag, 1968. | MR 233275 | Zbl 0159.50001
[36] R. H. F. DENNISTON, Some maximal arcs in finite projective planes, J. Combinatorial Theory, 6 (1969), 317-319. | MR 239991 | Zbl 0167.49106
[37] G. FELLEGARA, 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. | MR 155224 | Zbl 0109.39403
[38] R. A. FISHER, The theory of confounding in factorial experiments in relation to the theory of groups, Ann. Eugen. London, 11 (1942), 341-352. | MR 7382 | Zbl 0060.31502
[39] R. A. FISHER, A system confounding for factors with more than two alternatives giving completely orthogonal cubes and higher powers, Ann. Eugen. London, 12 (1945), 283-290. | MR 13113 | Zbl 0060.31503
[40] D. G. GLYNN, 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. | MR 731584 | Zbl 0531.51010
[41] R. W. HAMMING, Error detecting and error correcting codes, Bell Syst. Tech. J., 29 (1950), 147-160. | MR 35935
[42] R. HILL, On the larget size of cap in $S_{5,3}$, Atti Accad. Naz. Lincei, Rend. Cl. Sc. Mat., Fis., Nat., 54 (1973), 378-384. | MR 363951 | Zbl 0342.05017
[43] J. W. P. HIRSCHFELD, Projecttive geometries over finite fields, Oxford University Press, Oxford, 1979. | MR 554919 | Zbl 0418.51002
[44] J. W. P. HIRSCHFELD, Finite projective spaces of three dimensions, Clarendon Press, Oxford, 1985. | MR 840877 | Zbl 0574.51001
[45] J. W. P. HIRSCHFELD-J. A. THAS, General Galois Geometries, Oxford Mathematical Monographs, Oxford University Press, 1991. | MR 1363259 | Zbl 0789.51001
[46] D. R. HUGHES, On $t$-designs and groups, Amer. J. Math., 87 (1965), 761-768. | MR 188288 | Zbl 0134.03004
[47] D. R. HUGHES-F. C. PIPER, Projective planes, Springer Verlag, 1973. | MR 333959 | Zbl 0267.50018
[48] D. R. HUGHES-F. C. PIPER, Design Theory, Cambridge University Press, 1988. | MR 938742 | Zbl 0659.05008
[49] R. JAMISON, Covering finite fields with cosets of subspaces, J. of Combin. Theory (A), 22 (1977), 253-266. | MR 439664 | Zbl 0354.12019
[50] G. JÄRNEFELT-P. KUSTAANHEIMO, An observation on finite geometries, Comptes Rendus XI Congr. Math. Scand., Trodheim, 166-182, 1949. | MR 54979 | Zbl 0048.37103
[51] G. KORCHMAROS-F. MAZZOCCA, On $(q+t)$-arcs of type $(0, 2, t)$ in a Desarguesian plane of order $q\langle i \rangle$, Mathematical Proceedings of the Cambridge Philosophical Society, 108 (1990), 445-449. | MR 1068447
[52] C. W. H. LAM-L. THIEL-S. SWIERCZ-J. MCKAY, The non-existence of ovals in a projective plane of order 10, Discrete Math., 45 (1983), 319-321. | MR 704249 | Zbl 0574.05008
[53] C. W. H. LAM-L. THIEL-S. SWIERCZ, The non-existence of finite projective planes of order 10, Canad. J. Math., 41 (1989), 1117-1123. | MR 1018454 | Zbl 0691.51003
[54] L. LUNELLI-M. SCE, $k$-Archi completi nei piani proiettivi desarguesiani di rango 8 e 16. Centro di Calcoli numerici, Politecnico di Milano, 1958. | MR 157276
[55] F. MAZZOCCA, Blocking sets with respect to special families of lines and nuclei of $\theta_n$-sets in finite $n$-dimensional projective and affine spaces, Mitt. Math. Sem. Giessen, 201 (1991), 109-117. | MR 1126310 | Zbl 0741.51014
[56] J. VON NEUMANN-O. MORGENSTERN, Theory of games and economic behavior, Princeton, 1947. | MR 21298 | Zbl 0053.09303
[57] G. PANELLA, Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito, Boll. U.M.I., 10, Serie III (1955), 507-513. | fulltext bdim | fulltext mini-dml | MR 75607 | Zbl 0066.38902
[58] J. DI PAOLA, On a restricted class of block design games, Canadian J. Math., 18 (1966), 225-236. | MR 194207 | Zbl 0228.05013
[59] J. DI PAOLA, On minimum blocking coalitions in small projective games, SIAM J. of Appl. Math., 17 (1969), 378-392. | MR 247879 | Zbl 0191.49601
[60] S. E. PAYNE, A new infinite family of generalized quadrangles, Congr. Numer., 49 (1985), 115-128. | MR 830735 | Zbl 0632.51011
[61] G. PELLEGRINO, Sul massimo ordine delle calotte in $S_{4,3}$, Matematiche (Catania), 25 (1970), 1-9. | MR 363952
[62] T. PENTTILA-G. F. ROYLE, Sets of type $(m, n)$ in the affine and projective planes of order nine, Designs, Codes and Cryptography, 6 (1995), 229-245. | MR 1351846 | Zbl 0841.05012
[63] B. QVIST, Some remarks concerning curves of the second degree in a finite plane, Ann. Acad. Sci. Fenn., Ser. A, 134, 1952. | MR 54977 | Zbl 0049.10801
[64] L. RÉDEI, Lacunary polynomials over finite fields, North Holland, Amsterdam, 1973. | MR 352060 | Zbl 0255.12009
[65] M. RICHARDSON, On finite projective games, Proc. Amer. Math. Soc., 7 (1956), 458-465. | MR 79543 | Zbl 0073.35403
[66] M. SCAFATI-G. TALLINI, Geometria di Galois e teoria dei codici, CISU, Roma, 1995.
[67] B. SEGRE, Lezioni di geometria moderna, vol. I, Zanichelli, Bologna, 1948. | MR 30204 | Zbl 0030.41005
[68] B. SEGRE, Sulle ovali dei piani lineari finiti, Atti Acc. Naz. Lincei, Rend. Cl. Sci. Mat., Fis., Nat., 17 (1954), 141-142. | MR 71035 | Zbl 0057.36301
[69] B. SEGRE, Ovals in finite projective planes, Canad. J. Math., 7 (1955), 414-416. | MR 71034 | Zbl 0065.13402
[70] B. SEGRE, Sui $k$-archi nei piani finiti di caratteristica due, Rev. Math. Pures Appl., 2 (1957), 289-300. | MR 98347 | Zbl 0087.35003
[71] B. SEGRE, On complete caps and ovaloids in three dimensional Galois spaces of characteristic two, Acta Arith., 5 (1959), 315-332. | fulltext mini-dml | MR 114153 | Zbl 0094.15902
[72] B. SEGRE, Le geometrie di Galois, Ann. Mat. Pura e Appl., 48 (1959), 1-97. | MR 116259 | Zbl 0093.33604
[73] B. SEGRE, Lectures on modern geometry, Ed. Cremonese, Roma, 1961. | MR 131192 | Zbl 0095.14802
[74] B. SEGRE, Ovali e curve $\sigma$ nei piani di Galois di caratteristica due, Atti Acc. Naz. Lincei, Rend. Cl. Sci. Mat., Fis., Nat., 33 (1962), 785-790. | MR 149361 | Zbl 0107.38102
[75] B. SEGRE-U. BARTOCCI, Ovali ed altre curve nei piani di Galois di caratteristica due, Acta Arith., 18 (1971), 423-449. | fulltext mini-dml | MR 295201 | Zbl 0219.50016
[76] B. SEGRE-G. KORCHMAROS, 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. | MR 513866 | Zbl 0383.51006
[77] G. TALLINI, Sulle $k$-calotte di uno spazio lineare finito, Ann. Mat., 42 (1956), 119-164. | MR 85531 | Zbl 0074.15304
[78] G. TALLINI, Partial line spaces and algebraic varieties, Symposia Mathematica, XXVIII (1986), 203-217. | MR 873781 | Zbl 0616.51019
[79] G. TALLINI, 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] J. A. THAS, 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. | Zbl 0232.50013
[81] J. A. THAS, 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. | Zbl 0311.50016
[82] J. TITS, Les groupes simples de Suzuki et de Ree, Seminaire Bourbaki, 210, 1960. | fulltext mini-dml | MR 1611778 | Zbl 0267.20041
[83] J. TITS, Ovoides et groupes du Suzuki, Arch. Math., 13 (1962), 187-198. | MR 140572 | Zbl 0109.39402
[84] J. TITS, Un proprieté caracteristique des ovoïdes associés aux groupes de Suzuki, Arch. Math., 17 (1966), 136-153. | MR 203564 | Zbl 0136.15701
[85] V. D. TONCHEV, Combinatorial Configurations, Longman Scientific & Technical, 1988. | Zbl 0545.05002

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