Alcuni problemi di ricerca nelle geometrie di Galois
Bollettino della Unione matematica italiana, Série 8, 4B (2001) no. 1, pp. 45-69.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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.
@article{BUMI_2001_8_4B_1_a10,
     author = {Mazzocca, Francesco},
     title = {Alcuni problemi di ricerca nelle geometrie di {Galois}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {45--69},
     publisher = {mathdoc},
     volume = {Ser. 8, 4B},
     number = {1},
     year = {2001},
     zbl = {1036.51008},
     mrnumber = {82463},
     language = {it},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2001_8_4B_1_a10/}
}
TY  - JOUR
AU  - Mazzocca, Francesco
TI  - Alcuni problemi di ricerca nelle geometrie di Galois
JO  - Bollettino della Unione matematica italiana
PY  - 2001
SP  - 45
EP  - 69
VL  - 4B
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2001_8_4B_1_a10/
LA  - it
ID  - BUMI_2001_8_4B_1_a10
ER  - 
%0 Journal Article
%A Mazzocca, Francesco
%T Alcuni problemi di ricerca nelle geometrie di Galois
%J Bollettino della Unione matematica italiana
%D 2001
%P 45-69
%V 4B
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2001_8_4B_1_a10/
%G it
%F BUMI_2001_8_4B_1_a10
Mazzocca, Francesco. Alcuni problemi di ricerca nelle geometrie di Galois. Bollettino della Unione matematica italiana, Série 8, 4B (2001) no. 1, pp. 45-69. http://geodesic.mathdoc.fr/item/BUMI_2001_8_4B_1_a10/

[1] E. Artin, Geometric Algebra, Interscience, 1957. | MR | Zbl

[2] E. F. Assmus-E. F. Key, Designs and their Codes, Cambridge Tracts in Mathematics, 103, Cambridge University Press, 1992. | MR | Zbl

[3] S. Ball-A. Blokhuis-F. Mazzocca, Maximal arcs in Desarguesian planes of odd order do not exist, Combinatorics, 17 (1997), 31-41. | MR | Zbl

[4] A. Barlotti, Un'estensione del teorema di Segre-Kustaanheimo, Boll. U.M.I., 10, Serie III (1955), 498-506. | fulltext bdim | MR | Zbl

[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 | Zbl

[7] A. Beutelspacher, Einfurung in die endliche geometrie, vol. I, II, Wiessensnaftsverlag, 1983. | MR | Zbl

[8] A. Beutelspacher-U. Rosenbaum, Projective Geometry, Cambridge University Press, 1998. | MR | Zbl

[9] N. L. Biggs-A. T. White, Permutation Groups and Combinatorial Structures, London Mathematical Society, Lecture Note Series, 33, Cambridge University Press, 1979. | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[13] A. Blokhuis-F. Mazzocca, Lifts of nuclei in finite projective spaces, L.M.S. Lecture Note Series, 191 (1993), 31-36. | MR | Zbl

[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 | Zbl

[15] R. C. Bose, Mathematical Theory of the Symmetrical Factorial Design, Sankhya, 8 (1947), 107-166. | MR | Zbl

[16] R. C. Bose, On some connections between the design of experiments and information theory, Bull. Inst. Internat. Statist., 38 (1964), 257-271. | MR | Zbl

[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 | Zbl

[18] A. E. Brouwer-A. Schrijver, The blocking number of an affine space, J. of Combin. Theory (A), 24 (1978), 251-253. | MR | Zbl

[19] M. R. Brown, Ovoids of $PG(3, q)$, $q$ even, with a conic section, J. London Math. Soc., to appear. | MR

[20] A. A. Bruen, Baer subplanes and blocking sets, Bull. Amer. Math. Soc., 76 (1970), 342-344. | fulltext mini-dml | MR | Zbl

[21] A. A. Bruen, Blocking sets in finite projective planes, SIAM J. of Appl. Math., 21 (1971), 380-392. | MR | Zbl

[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 | Zbl

[23] A. A. Bruen-M. J. De Resmini, Blocking sets in affine planes, Annals of Discrete Math., 18 (1983), 169-176. | MR | Zbl

[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 | Zbl

[25] A. A. Bruen-J. A. Thas, Blocking sets, Geom. Ded., 6 (1977), 193-203. | MR | Zbl

[26] P. J. Cameron, Four lectures on projective geometry, in Finite Geometries, Lecture Notes in Pure and Applied Mathematics, 103 (1985), 27-65. | MR | Zbl

[27] P. J. Cameron-J. H. Van Lint, Designs, Graphs, Codes and their Links, London Mathematical Society, Student Texts, 22, 1991. | MR | Zbl

[28] P. J. Cameron-F. Mazzocca, Bijections which preserve blocking sets, Geom. Dedicata, 21 (1986), 219-229. | MR | Zbl

[29] M. Cerasoli-F. Eugeni-M. Protasi, Elementi di matematica discreta, Zanichelli, 1988. | MR | Zbl

[30] W. Cherowitzo, Hyperovals in Desarguesian planes of even order, Ann. Discrete Math., 37 (1988), 87-94. | MR | Zbl

[31] W. Cherowitzo-T. Penttila-I. Pinneri, Flocks and ovals, Geom. Dedicata, 60 (1996), 17-37. | MR | Zbl

[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 | Zbl

[34] H. S. M. Coxeter, Projective geometry, Springer, Second Edition, 1987. | MR | Zbl

[35] P. Dembowski, Finite geometries, Springer-Verlag, 1968. | MR | Zbl

[36] R. H. F. Denniston, Some maximal arcs in finite projective planes, J. Combinatorial Theory, 6 (1969), 317-319. | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[41] R. W. Hamming, Error detecting and error correcting codes, Bell Syst. Tech. J., 29 (1950), 147-160. | MR

[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 | Zbl

[43] J. W. P. Hirschfeld, Projecttive geometries over finite fields, Oxford University Press, Oxford, 1979. | MR | Zbl

[44] J. W. P. Hirschfeld, Finite projective spaces of three dimensions, Clarendon Press, Oxford, 1985. | MR | Zbl

[45] J. W. P. Hirschfeld-J. A. Thas, General Galois Geometries, Oxford Mathematical Monographs, Oxford University Press, 1991. | MR | Zbl

[46] D. R. Hughes, On $t$-designs and groups, Amer. J. Math., 87 (1965), 761-768. | MR | Zbl

[47] D. R. Hughes-F. C. Piper, Projective planes, Springer Verlag, 1973. | MR | Zbl

[48] D. R. Hughes-F. C. Piper, Design Theory, Cambridge University Press, 1988. | MR | Zbl

[49] R. Jamison, Covering finite fields with cosets of subspaces, J. of Combin. Theory (A), 22 (1977), 253-266. | MR | Zbl

[50] G. Järnefelt-P. Kustaanheimo, An observation on finite geometries, Comptes Rendus XI Congr. Math. Scand., Trodheim, 166-182, 1949. | MR | Zbl

[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

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

[56] J. Von Neumann-O. Morgenstern, Theory of games and economic behavior, Princeton, 1947. | MR | Zbl

[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 | Zbl

[58] J. Di Paola, On a restricted class of block design games, Canadian J. Math., 18 (1966), 225-236. | MR | Zbl

[59] J. Di Paola, On minimum blocking coalitions in small projective games, SIAM J. of Appl. Math., 17 (1969), 378-392. | MR | Zbl

[60] S. E. Payne, A new infinite family of generalized quadrangles, Congr. Numer., 49 (1985), 115-128. | MR | Zbl

[61] G. Pellegrino, Sul massimo ordine delle calotte in $S_{4,3}$, Matematiche (Catania), 25 (1970), 1-9. | MR

[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 | Zbl

[63] B. Qvist, Some remarks concerning curves of the second degree in a finite plane, Ann. Acad. Sci. Fenn., Ser. A, 134, 1952. | MR | Zbl

[64] L. Rédei, Lacunary polynomials over finite fields, North Holland, Amsterdam, 1973. | MR | Zbl

[65] M. Richardson, On finite projective games, Proc. Amer. Math. Soc., 7 (1956), 458-465. | MR | Zbl

[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 | Zbl

[68] B. Segre, Sulle ovali dei piani lineari finiti, Atti Acc. Naz. Lincei, Rend. Cl. Sci. Mat., Fis., Nat., 17 (1954), 141-142. | MR | Zbl

[69] B. Segre, Ovals in finite projective planes, Canad. J. Math., 7 (1955), 414-416. | MR | Zbl

[70] B. Segre, Sui $k$-archi nei piani finiti di caratteristica due, Rev. Math. Pures Appl., 2 (1957), 289-300. | MR | Zbl

[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 | Zbl

[72] B. Segre, Le geometrie di Galois, Ann. Mat. Pura e Appl., 48 (1959), 1-97. | MR | Zbl

[73] B. Segre, Lectures on modern geometry, Ed. Cremonese, Roma, 1961. | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[77] G. Tallini, Sulle $k$-calotte di uno spazio lineare finito, Ann. Mat., 42 (1956), 119-164. | MR | Zbl

[78] G. Tallini, Partial line spaces and algebraic varieties, Symposia Mathematica, XXVIII (1986), 203-217. | MR | Zbl

[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

[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

[82] J. Tits, Les groupes simples de Suzuki et de Ree, Seminaire Bourbaki, 210, 1960. | fulltext mini-dml | MR | Zbl

[83] J. Tits, Ovoides et groupes du Suzuki, Arch. Math., 13 (1962), 187-198. | MR | Zbl

[84] J. Tits, Un proprieté caracteristique des ovoïdes associés aux groupes de Suzuki, Arch. Math., 17 (1966), 136-153. | MR | Zbl

[85] V. D. Tonchev, Combinatorial Configurations, Longman Scientific & Technical, 1988. | Zbl