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4 - Combinatorial structures in finite classical polar spaces

Published online by Cambridge University Press:  21 July 2017

By
Antonio Cossidente
Edited by
Anders Claesson ,
Mark Dukes ,
Sergey Kitaev ,
David Manlove  and
Kitty Meeks
Antonio Cossidente
Affiliation:
Dipartimento di Matematica Informatica ed Economia
Anders Claesson
Affiliation:
University of Iceland, Reykjavik
Mark Dukes
Affiliation:
University College Dublin
Sergey Kitaev
Affiliation:
University of Strathclyde
David Manlove
Affiliation:
University of Glasgow
Kitty Meeks
Affiliation:
University of Glasgow
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Summary

Abstract

Sabsome recent results on regular systems and intriguing sets of finite classical polar spaces are surveyed.

Introduction

The finite classical polar spaces are the geometries associated with non–degenerate reflexive sesquilinear and non–singular quadratic forms on vector spaces of finite dimension over a finite field GF(q). Let PG(n, q) denote the n–dimensional projective space over GF(q). A polar space in PG(n, q) consists of its projective subspaces that are totally isotropic with respect to a given non–degenerate reflexive sesquilinear form or that are totally singular with respect to a given non–singular quadratic form. The rank of a polar space is the vector space dimension of a maximal totally isotropic or totally singular subspace, called here maximal. In this paper, the term polar space always refers to a finite classical polar space. A polar space of rank two is a generalised quadrangle. In the last decades, intensive investigations on combinatorial structures in finite polar spaces, such as spreads, ovoids, blocking sets, covers, have been carried out. More recently, other structures, such as m–systems, m–ovoids, i–tight sets (intriguing sets) have been studied. In this paper, some recent results on regular systems and intriguing sets of finite polar spaces are surveyed, with special emphasis on Hermitian polar spaces, that is, polar spaces arising from a non–degenerate unitary form.

Throughout the paper the following notation is adopted for finite polar spaces:

  1. (1)H(n, q2) is the space associated with a non–degenerate hermitian form on a vector space of dimension n + 1 over GF(q2);

  2. (2)Q-(n, q) is the space associated with a non –singular quadratic form of non–maximal Witt index on a vector space of dimension n+1 even over GF(q);

  3. (3)Q+(n, q) is the space associated with a non–singular quadratic form of maximal Witt index on a vector space of even dimension n+1 over GF(q);

  4. (4)Q(n, q) is the space associated with a non–singular quadratic form on a vector space of odd dimension n + 1 over GF(q);

  5. (5)W(n, q) is the space associated with a non–degenerate symplectic form on a vector space of even dimension n + 1 over GF(q).

Type
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Surveys in Combinatorics 2017 , pp. 204 - 237
Publisher: Cambridge University Press
Print publication year: 2017

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References

[1] J., Bamberg, M., Lee, A relative m–cover of a Hermitian surface is a relative hemisystem, arXiv 1608.03055v1.
[2] Laura, Bader, Guglielmo, Lunardon, and Joseph A., Thas, Derivation of flocks of quadratic cones, Forum Math. 2 1990, 163–174.Google Scholar
[3] Simeon, Ball, Aart, Blokhuis, and Francesco, Mazzocca, Maximal arcs in Desarguesian planes of odd order do not exist, Combinatorica 17 1997, 31–41.Google Scholar
[4] John, Bamberg, Michael, Giudici, and Gordon F., Royle, Every flock generalized quadrangle has a hemisystem, Bull. London Math. Soc. 42 2010, 795–810.Google Scholar
[5] John, Bamberg, Michael, Giudici, and Gordon F., Royle, Hemisystems of small flock generalized quadrangles, Des. Codes Cryptogr. 67 2013, 137–157.Google Scholar
[6] John, Bamberg, Shane, Kelly, Maska, Law, and Tim, Penttila, Tight sets and m-ovoids of finite polar spaces, J. Combin. Theory Ser. A 114 2007, 1293–1314.Google Scholar
[7] John, Bamberg, Mashka, Law, and Tim, Penttila, Tight sets and movoids of finite generalised quadrangles, Combinatorica 29 2009, 1– 17.Google Scholar
[8] John, Bamberg, Melissa, Lee, and Eric, Swartz, A note on relative hemisystems of Hermitian generalized quadrangles, Des. Codes Cryptogr. (2015), to appear.
[9] John, Bamberg, Melissa, Lee, Koji, Momihara, and Qing, Xiang, A new infinite family of hemisystems of the Hermitian surface, preprint.
[10] Luke, Bayens, Hyperovals, Laguerre planes and hemisystems – An approach via symmetry, Ph.D. Thesis, Colorado State University, Fort Collins, Colorado, 2013.
[11] Antonia W., Bluher, On xq+1 +ax+b, Finite Fields Appl. 10 2004, 285–305.Google Scholar
[12] Ray C., Bose and T., Shimamoto, Classification and analysis of partially balanced incomplete block designs with two associate classes, J. Amer. Statist. Assoc. 47 1952, 151–184.Google Scholar
[13] John, Bray, Derek, Holt, and Colva, Roney-Dougal, The maximal subgroups of the low–dimensional finite classical groups, London Mathematical Society, LNS 407, Cambridge University Press, Cambridge, 2013.
[14] Aiden A., Bruen and James W.P., Hirschfeld, Applications of line geometry over finite fields. II. The Hermitian surface, Geom. Dedicata 7 1978, 333–353.Google Scholar
[15] Aiden A., Bruen and Keldon Drudge, The construction of Cameron- Liebler line classes in PG(3, q), Finite Fields Appl. 5 1999, 35–45.Google Scholar
[16] Peter J., Cameron, Partial quadrangles, Quart. J. Math. Oxford Ser. 26 1975, 61–73.Google Scholar
[17] Peter J., Cameron, Philippe, Delsarte, and Jean-Marie, Goethals, Hemisystems, orthogonal configurations and dissipative conference matrices, Philips J. Res. 34 1979, 147–162.Google Scholar
[18] Peter J., Cameron, Jean-Marie, Goethals, and Johan J., Seidel, Strongly regular graphs having strongly regular subconstituents, J. Algebra 55 1978, 257–280.Google Scholar
[19] Peter J., Cameron and Robert A., Liebler, Tactical decompositions and orbits of projective groups, Linear Algebra Appl. 46 1982, 91–102.Google Scholar
[20] John, Cannon and Catherine, Playoust, An introduction to MAGMA, University of Sydney, Sydney, Australia, 1993.
[21] Miroslava, Cimráková and Veerle, Fack, Searching for maximal partial ovoids and spreads in generalized quadrangles, Bull. Belg. Math. Soc. Simon Stevin 12 2005, 697–705.Google Scholar
[22] Miroslava, Cimráková and Veerle, Fack, Clique algorithms for finding substructures in generalized quadrangles, J. Combin. Math. Combin. Comput. 63 2007, 129–143.Google Scholar
[23] Antonio, Cossidente, Relative hemisystems on the Hermitian surface, J. Algebraic Combin. 38 2013, 275–284.Google Scholar
[24] Antonio, Cossidente, A new family of relative hemisystems on the Hermitian surface, Des. Codes Cryptogr. 75 2005, 213–221.Google Scholar
[25] Antonio, Cossidente and Gary L., Ebert, Permutable polarities and a class of ovoids of the Hermitian surface, European J. Combin. 25 (2004), 1059–1066.Google Scholar
[26] Antonio, Cossidente, Gabor, Korchmáros, and Francesco, Pavese, A new approach to relative hemisystems, preprint (2016).
[27] Antonio, Cossidente, Giuseppe, Marino, and Olga, Polverino, Special sets of the Hermitian surface and indicator sets, J. Combin. Des. 16 2008, 18–24.Google Scholar
[28] Antonio, Cossidente and Francesco, Pavese, Intriguing sets of quadrics in PG(5, q), Adv. Geom., to appear.
[29] Antonio, Cossidente and Francesco, Pavese, On the geometry of unitary involutions, Finite Fields Appl. 36 2015, 14–28.Google Scholar
[30] Antonio, Cossidente and Francesco, Pavese, Hemisystems of Q(6, q), q odd, J. Combin. Theory Ser. A, to appear.
[31] Antonio, Cossidente and Francesco, Pavese, Intriguing sets of W(5, q), q even, J. Combin Theory Ser. A 127 2014, 303–313.Google Scholar
[32] Antonio, Cossidente and Francesco, Pavese, Small tight sets of W(4n﹜ 1, q2), preprint.
[33] Antonio, Cossidente and Francesco, Pavese, Relative m–ovoids of elliptic quadrics, submitted.
[34] Antonio, Cossidente and Tim, Penttila, Hemisystems on the Hermitian surface, J. London Math. Soc. 72 2005, 731–741.Google Scholar
[35] Antonio, Cossidente and Tim, Penttila, On m-regular systems on H(5, q2), J. Algebraic Combin. 29 2009, 437–445.Google Scholar
[36] Antonio, Cossidente and Tim, Penttila, Segre's hemisystem and McLaughlin's graph, J. Combin. Theory Ser. A 115 2008, 686–692.Google Scholar
[37] Antonio, Cossidente and Tim, Penttila, Subquadrangle m-regular systems on generalized quadrangles, J. Combin. Des. 19 2011, 28–41.Google Scholar
[38] Antonio, Cossidente and Hendrik Van, Maldeghem, The simple exceptional group G2(q), q even, and two-character sets, J. Combin. Theory Ser. A 114 2007, 964–969.Google Scholar
[39] Antonio, Cossidente and Sam K.J., Vereecke, Some geometry of the isomorphism Sp(4, q) ≃O(5, q), q even, J. Geom. 70 2001, 28–37.Google Scholar
[40] Jan De, Beule, Patrick, Govaerts, Anja, Hallez, and Leo, Storme, Tight sets, weighted m–covers, weighted m–ovoids, and minihypers, Des. Codes Cryptogr. 50 2009, 187–201.Google Scholar
[41] Franck De, Clerck, Nikias De, Feyter, Nicola, Durante, Two-intersection sets with respect to lines on the Klein quadric, Bull. Belg. Math. Soc. Simon Stevin 12 2005, 743–750.Google Scholar
[42] Keldon, Drudge, Extremal sets in projective and polar spaces, Ph. D. Thesis, University of Western Ontario, Canada, 1998.
[43] Roger H., Dye, Spreads and classes of maximal subgroups of GLn(q), SLn(q),PGLn(q) and PSLn(q), Ann. Mat. Pura Appl. 158 1991, 33–50.Google Scholar
[44] Joergen, Eisfeld, On the common nature of spreads and pencils in PG(d, q), Discrete Math. 189 1998, 95–104.Google Scholar
[45] Tao, Feng, Koji, Momihara, and Qing, Xiang, Cameron–Liebler line classes with parameter x = (q2 -1)/2, J. Combin. Theory Ser. A 133 2015, 307–338.Google Scholar
[46] Patrick, Govaerts and Tim, Penttila, Cameron-Liebler line classes in PG(3, 4), Bull. Belg. Math. Soc. Simon Stevin 12 2005, 793–804.Google Scholar
[47] James W.P., Hirschfeld, Finite projective spaces of three dimensions, Oxford Mathematical Monographs, Oxford Science Publications,The Clarendon Press, Oxford University Press, Oxford, 1985.
[48] Barbu C., Kestenband, Projective geometries that are disjoint unions of caps, Canad. J. Math., 32 1980, 1299–1305.Google Scholar
[49] Peter, Kleidman and Martin, Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge University Press, Cambridge (1990).
[50] Norbert, Knarr, A geometric construction of generalized quadrangles from polar spaces of rank three, Results Math. 21 1992, 332–344.Google Scholar
[51] Michel, Lavrauw and Gertrude Van de, Voorde, Field reduction and linear sets in finite geometry, Topics in finite fields, 271–293, Contemp. Math., 632, Amer. Math. Soc., Providence, RI, 2015.
[52] Deirdre, Luyckx, m–systems of finite classical polar spaces, Ph.D. Thesis, Ghent University (2002).
[53] Deirdre, Luyckx and Joseph A., Thas, The uniqueness of the 1-system of Q-(7, q), q odd, J. Combin. Theory Ser. A 98 2002, 253–267.Google Scholar
[54] William J., Martin, Mikhail, Muzychuk, and Jason, Williford, Imprimitive cometric association schemes: constructions and analysis, J. Algebraic Combin. 25 2007, 399–415.Google Scholar
[55] Stanley E., Payne and Joseph A., Thas, Finite Generalized Quadrangles, Res. Notes Math. 110, Pitman, 1984.
[56] Tim, Penttila, Cameron–Liebler line classes in PG(3, q), Geom. Dedicata 37 1991, 245–252.Google Scholar
[57] Tim, Penttila and J., Williford, New families of Q-polynomial association schemes, J. Combin. Theory Ser. A 118 2011, 502–509.Google Scholar
[58] Joergen, Rathmann, The uniform principle for curves in characteristic p, Math Ann. 276 1987, 565–579.Google Scholar
[59] Beniamino, Segre, Forme e geometrie hermitiane, con particolare riguardo al caso finito, Ann. Mat. Pura Appl. 70 1965, 1–201.Google Scholar
[60] Beniamino, Segre, Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl. 64 1964, 1–76.Google Scholar
[61] Ernest E., Shult and Joseph A., Thas, Construction of polygons from buildings, Proc. London Math. Soc. 71 1995, 397–440.Google Scholar
[62] Ernest E., Shult and Joseph A., Thas, m-systems of polar spaces, J. Combin. Theory Ser. A 68 1994, 184–204.Google Scholar
[63] Joseph A., Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 1981, 135–143.Google Scholar
[64] Joseph A., Thas, Projective geometry over a finite field, Handbook of Incidence Geometry, (Buekenhout, F., ed.), 295–347, North-Holland, Amsterdam, 1995.
[65] Joseph A., Thas, Interesting pointsets in generalized quadrangles and partial geometries, Linear Algebra Appl. 114/115 (1989), 103–131.Google Scholar
[66] Hendrik, VanMaldeghem, Generalized polygons, Monographs in Mathematics, vol. 93, Birkhäuser, Basel, (1998).

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