Abstract
Some constructions of intriguing sets of finite symplectic spaces are provided. In particular an affirmative answer to an existence question about small tight sets posed in De Beule et al. (Des Codes Cryptogr 50(2):187–201, 2009) is given.
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Bamberg J., Kelly S., Law M., Penttila T.: Tight sets and \(m\)-ovoids of finite polar spaces. J. Combin. Theory Ser. A 114(7), 1293–1314 (2007).
Bray J., Holt D., Roney-Dougal C.: The Maximal Subgroups of the Low-Dimensional Finite Classical Groups. London Mathematical Society, LNS 407, Cambridge University Press, New York (2013).
Bruen A.A., Hirschfeld J.W.P.: Intersections in projective space. I. Combinatorics. Math. Z. 193(2), 215–225 (1986).
Butler D.K.: On the intersection of ovoids sharing a polarity. Geom. Dedicata 135, 157–165 (2008).
Cossidente A.: On Kestenband–Ebert partitions. J. Combin. Des. 5(5), 367–375 (1997).
Cossidente A., Culbert C., Ebert G.L., Marino G.: On \(m\)-ovoids of \({\cal{W}}(3, q)\). Finite Fields Appl. 14(1), 76–84 (2008).
Cossidente A., Pavese F.: On the geometry of unitary involutions. Finite Fields Appl. 36, 14–28 (2015).
Cossidente A., Pavese F.: Subspace codes in \({\rm PG}(2n-1,q)\). Combinatorica. doi:10.1007/s00493-016-3354-5.
Cossidente A., Pavese F.: Intriguing sets of \({\cal{W}}(5, q)\), \(q\) even. J. Combin. Theory Ser. A 127, 303–313 (2014).
De Beule J., Govaerts P., Hallez A., Storme L.: Tight sets, weighted \(m\)-covers, weighted \(m\)-ovoids, and minihypers. Des. Codes Cryptogr. 50(2), 187–201 (2009).
Dye R.H.: Spreads and classes of maximal subgroups of \({\rm GL}_n(q),{\rm SL}_n(q),{\rm PGL}_n(q)\) and \({\rm PSL}_n(q)\). Ann. Mat. Pura Appl. 158(4), 33–50 (1991).
Dye R.H.: Partitions and their stabilizers for line complexes and quadrics. Ann. Mat. Pura Appl. (V) 114, 173–194 (1977).
Drudge K.: Proper \(2\)-covers of \({\rm PG}(3, q)\), \(q\) even. Geom. Dedicata 80(1–3), 59–64 (2000).
Ebert G.L.: Partitioning projective geometries into caps. Can. J. Math. 37(6), 1163–1175 (1985).
Hirschfeld J.W.P.: Projective Geometries over Finite Fields. Oxford Mathematical MonographsOxford Science Publications, The Clarendon Press, Oxford University Press, New York (1998).
Kelly S.: Constructions of intriguing sets of polar spaces from field reduction and derivation. Des. Codes Cryptogr. 43(1), 1–8 (2007).
Kestenband B.C.: Projective geometries that are disjoint unions of caps. Can. J. Math. 32(6), 1299–1305 (1980).
Kleidman P., Liebeck M.: The Subgroup Structure of the Finite Classical Groups, vol. 129. London Mathematical Society Lecture Note SeriesCambridge University Press, Cambridge (1990).
Nakić A., Storme L.: Tight sets in finite classical polar spaces. Adv. Geom. 17(1), 109–129 (2017).
O’Keefe C.M.: Ovoids in \({\rm PG}(3, q)\): a survey. Discret. Math. 151, 171–188 (1996).
Pavese F.: Geometric constructions of two-character sets. Discret. Math. 338(3), 202–208 (2015).
Segre B.: Forme e geometrie hermitiane, con particolare riguardo al caso finito. Ann. Mat. Pura Appl. 70(4), 1–201 (1965).
Segre B.: On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two. Acta Arith. 5, 315–332 (1959).
Segre B.: Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane. Ann. Mat. Pura Appl. 64, 1–76 (1964).
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Communicated by J. W. P. Hirschfeld.
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Cossidente, A., Pavese, F. On intriguing sets of finite symplectic spaces. Des. Codes Cryptogr. 86, 1161–1174 (2018). https://doi.org/10.1007/s10623-017-0387-8
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DOI: https://doi.org/10.1007/s10623-017-0387-8