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Recent results on designs with classical parameters

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We report on recent results concerning designs with the same parameters as the classical geometric designs PG d (n, q) formed by the points and d-dimensional subspaces of the n-dimensional projective space PG(n, q) over the field GF(q) with q elements, where 1 ≤ d ≤ n−1. The corresponding case of designs with the same parameters as the classical geometric designs AG d (n, q) formed by the points and d-dimensional subspaces of the n-dimensional affine space AG(n, q) will also be discussed, albeit in less detail.

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References

  1. Assmus E.F. Jr., Key J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  2. Baartmans A., Sane S.: A characterization of projective subspaces of codimension two as quasi-symmetric designs with good blocks. Discrete Math. 306, 1493–1501 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beth T., Jungnickel D., Lenz H.: Design Theory, 2nd edn. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  4. Brouwer A.E., Cohen A.M., Neumaier A.: Distance Regular Graphs. Springer, Berlin (1989)

    MATH  Google Scholar 

  5. Clark D., Jungnickel D., Tonchev V.D.: Affine geometry designs, polarities, and Hamada’s conjecture. J. Combin. Theory Ser. A. 118, 231–239 (2011). doi:10.1016/j.jcta.2010.06.007

    Article  MathSciNet  MATH  Google Scholar 

  6. Clark D., Jungnickel D., Tonchev V.D.: Exponential bounds on the number of designs with affine parameters. J. Combin. Des. 19, 156–166 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Colbourn C.J., Dinitz J.H.: The CRC Handbook of Combinatorial Designs, 2nd edn. Chapman & Hall/CRC, Boca Raton (2007)

    MATH  Google Scholar 

  8. van Dam E.R., Koolen J.H.: A new family of distance-regular graphs with unbounded diameter. Invent. Math. 162, 189–193 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dehon M.: Designs et hyperplans. J. Combin. Theory Ser. A 23, 264–274 (1977)

    Article  MathSciNet  Google Scholar 

  10. Dembowski P.: Eine Kennzeichnung der endlichen affinen Räume. Arch. Math. 15, 146–154 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dembowski, P.: Finite Geometries. Reprint. Springer, Berlin (1997)

  12. Dembowski P., Wagner A.: Some characterizations of finite projective spaces. Arch. Math. 11, 465–469 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dempwolff U., Kantor W.M.: Distorting symmetric designs. Des. Codes Cryptogr. 48, 307–322 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Donovan D.M., Grannell M.J.: Designs having the parameters of projective and affine spaces. Des. Codes Cryptogr. 60, 225–240 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Doyen J., Hubaut X., Vandensavel M.: Ranks of incidence matrices of Steiner triple systems. Math. Z. 163, 251–259 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fujisaki T., Koolen J.H., Tagami M.: Some properties of the twisted Grassmann graphs. Innov. Incidence Geom. 3, 81–87 (2006)

    MathSciNet  MATH  Google Scholar 

  17. Godsil C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993)

    MATH  Google Scholar 

  18. Hamada N.: The rank of the incidence matrix of points and d-flats in finite geometries. J. Sci. Hiroshima Univ. Ser. A-I 32, 381–396 (1968)

    MathSciNet  Google Scholar 

  19. Hamada N.: On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its application to error correcting codes. Hiroshima Math. J. 3, 154–226 (1973)

    MathSciNet  Google Scholar 

  20. Hamada N., Ohmori H.: On the BIB-design having the minimum p-rank. J. Combin. Theory Ser. A 18, 131–140 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  21. Harada M., Lam C.W.H., Tonchev V.D.: Symmetric (4,4)-nets and generalized Hadamard matrices over groups of order 4. Des. Codes Cryptogr. 34, 71–87 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  23. Hirschfeld J.W.P.: Finite Projective Spaces of Three Dimensions. Oxford University Press, Oxford (1985)

    MATH  Google Scholar 

  24. Hirschfeld, J.W.P., Shaw, R.: Projective geometry codes over prime fields. In: Finite Fields: Theory, Application and Algorithms. Contemporary Math., vol. 168, pp. 151–163. Amer Math. Soc., Providence (1994)

  25. Jensen J.L.W.V.: Sur une identité d’Abel et sur d’autres formules analogues. Acta Math. 26, 307–318 (1902)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jungnickel D.: The number of designs with classical parameters grows exponentially. Geom. Dedicata 16, 167–178 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jungnickel D.: Characterizing geometric designs. Rend. Mat. Appl. 30(7), 111–120 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Jungnickel D.: Characterizing geometric designs, II. J. Combin. Theory Ser. A. 118, 623–633 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jungnickel D., Tonchev V.D.: Polarities, quasi-symmetric designs, and Hamada’s conjecture. Des. Codes Cryptogr. 51, 131–140 (2009)

    Article  MathSciNet  Google Scholar 

  30. Jungnickel D., Tonchev V.D.: The number of designs with geometric parameters grows exponentially. Des. Codes Cryptogr. 55, 131–140 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kabler J., Schöning U., Torán J.: The Graph Isomorphism Problem: Its Structural Complexity. Birkhäuser, Basel (1993)

    Google Scholar 

  32. Kantor W.M.: Automorphisms and isomorphisms of symmetric and affine designs. J. Algebraic Combin. 3, 307–338 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lam C., Lam S., Tonchev V.D.: Bounds on the number of affine, symmetric, and Hadamard designs and matrices. J. Combin. Theory Ser. A 92, 186–196 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lam C., Tonchev V.D.: A new bound on the number of designs with classical affine parameters. Des. Codes Cryptogr. 27, 111–117 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Larsen, M.E.: Summa Summarum. CMS Treatises in Mathematics, Canadian Mathematical Society, Ottawa, ON; A K Peters, Ltd., Wellesley (2007)

  36. Lefèvre-Percsy C.: Characterizations of designs constructed from affine and projective spaces. Eur. J. Combin. 1, 347–352 (1980)

    MATH  Google Scholar 

  37. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

  38. McDonough T.P., Mavron V.C.: Quasi-symmetric designs with good blocks. J. Combin. Des. 3, 433–441 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  39. Metsch K.: A generalization of a result of Dembowski and Wagner. Des. Codes Cryptogr. Des. Codes Cryptogr. 60, 277–282 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Munemasa, A., Tonchev, V.D.: The twisted Grassmann graph is the block graph of a design. arXiv:0906.4509v2 [math.CO] (2009)

  41. Teirlinck L.: On projective and affine hyperplanes. J. Combin. Theory Ser. A 28, 290–306 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  42. Tonchev V.D.: Quasi-symmetric 2-(31, 7, 7)-designs and a revision of Hamada’s conjecture. J. Combin. Theory Ser. A 42, 104–110 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wilson R.M.: An existence theory for pairwise balanced designs, III. Proof of the existence conjectures. J. Combin. Theory (A) 18, 71–79 (1975)

    Article  MATH  Google Scholar 

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Correspondence to Dieter Jungnickel.

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Jungnickel, D. Recent results on designs with classical parameters. J. Geom. 101, 137–155 (2011). https://doi.org/10.1007/s00022-011-0086-y

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