Abstract
A new family of relative hemisystems on the generalized quadrangle \({\mathcal {H}}(3,q^2)\) admitting a group of order \(q^2(q+1)\) as automorphism group, is constructed. As a by product a new infinite family of hyperovals of \({\mathcal {H}}(3,q^2)\) is presented.
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Bruen A.A., Hirschfeld J.W.P.: Applications of line geometry over finite fields. II. The Hermitian surface. Geom. Dedicata 7(3), 333–353 (1978).
Bruen A.A., Hirschfeld J.W.P.: Intersections in projective space. II: pencils of quadrics. Eur. J. Comb. 9, 255–270 (1988).
Buekenhout F., Hubaut X.: Locally polar spaces and related rank \(3\) groups. J. Algebr. 45, 391–434 (1977).
Cameron P.J.: Partial quadrangles. Q. J. Math. Oxf. Ser. (2) 26, 61–73 (1975).
Cameron P.J., Goethals J.M., Seidel J.J.: Strongly regular graphs having strongly regular subconstituents. J. Algebr. 55(2), 257–280 (1978).
Cameron P.J., Hughes D.R., Pasini A.: Extended generalized quadrangles. Geom. Dedicata 35, 193–228 (1990).
Cossidente A.: Relative hemisystems on the Hermitian surface. J. Algebr. Comb. 38(2), 275–284 (2013).
Cossidente A., Penttila T.: Hemisystems on the Hermitian surface. J. Lond. Math. Soc. 72, 731–741 (2005).
Coxeter H.S.M.: Twelve points in \(PG(5,3)\) with \(95040\) self-transformations. Proc. R. Soc. Lond. Ser. A 247, 279–293 (1958).
De Bruyn B.: On hyperovals of polar spaces. Des. Codes Cryptogr. 56(2–3), 183–195 (2010).
Del Fra A., Ghinelli D., Payne S.E.: \((0, n)\)-sets in a generalized quadrangle, Combinatorics ’90 (Gaeta, 1990). Ann. Discret. Math. 52, 139–157 (1992).
Dembowski P.: Finite Geometries. Springer-Verlag, Berlin, New York (1968).
Dye R.H.: On the conjugacy classes of involutions of the unitary groups \(U_{m}(K), SU_{m}(K),\) \(PU_{m}(K),PSU_{m}(K)\), over perfect fields of characteristic \(2\). J. Algebr. 24, 453–459 (1973).
Haemers W.H., Higman D.G.: Strongly regular graphs with strongly regular decomposition. Linear Algebr. Appl. 114–115, 379–398 (1989).
Hill R.: Caps and groups. Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo II, pp. 389–394. Atti dei Convegni Lincei, No. 17, Accad. Naz. Lincei, Rome (1976).
Hirschfeld J.W.P.: Finite Projective Spaces of Three Dimensions. Oxford University Press, Oxford (1991).
Noda R.: Partitioning strongly regular graphs. Osaka J. Math. 22, 379–389 (1985).
Payne S.E., Thas J.A.: Finite Generalized Quadrangles. Research Notes in Mathematics, vol. 110. Pitman, Boston, MA (1984).
Penttila T., Williford J.: New families of \(Q\)-polynomial association schemes. J. Comb. Theory Ser. A 118, 502–509 (2011).
Segre B.: Forme e geometrie Hermitiane con particolare riguardo al caso finito. Ann. Mat. Pura Appl. 70, 1–201 (1965).
Thas J.A.: Ovoids and spreads of finite classical polar spaces. Geom. Dedicata 10, 135–143 (1981).
Thas J.A.: Projective geometry over a finite field. In: Buekenhout F. (ed.) Handbook of Incidence Geometry, pp. 295–347. North-Holland, Amsterdam (1995).
Thas K.: Symmetry in Finite Generalized Quadrangles. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004).
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Cossidente, A. A new family of relative hemisystems on the Hermitian surface. Des. Codes Cryptogr. 75, 213–221 (2015). https://doi.org/10.1007/s10623-013-9906-4
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DOI: https://doi.org/10.1007/s10623-013-9906-4