Advertisement

Designs, Codes and Cryptography

, Volume 68, Issue 1–3, pp 205–227 | Cite as

Finite semifields and nonsingular tensors

  • Michel Lavrauw
Article

Abstract

In this article, we give an overview of the classification results in the theory of finite semifields (note that this is not intended as a survey of finite semifields including a complete state of the art (see also Remark 1.10)) and elaborate on the approach using nonsingular tensors based on Liebler (Geom Dedicata 11(4):455–464, 1981).

Keywords

Finite semifields Projective planes Segre varieties Tensor products 

Mathematics Subject Classification

12K10 51A40 05E20 05B25 51E15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albert A.A.: Non-associative algebras. I. Fundamental concepts and isotopy. Ann. Math. (2) 43, 685–707 (1942)MATHCrossRefGoogle Scholar
  2. 2.
    Albert A.A.: On nonassociative division algebras. Trans. Am. Math. Soc. 72, 296–309 (1952)MATHCrossRefGoogle Scholar
  3. 3.
    Albert A.A.: Finite division algebras and finite planes. In: Proceedings of Symposia in Applied Mathematics, vol. 10, pp. 53–70. American Mathematical Society, Providence (1960).Google Scholar
  4. 4.
    Albert A.A.: Generalized twisted fields. Pac. J. Math. 11, 1–8 (1961)MATHCrossRefGoogle Scholar
  5. 5.
    André J.: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Math. Z. 60, 156–186 (1954)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Aschbacher M.: Isotopy and geotopy for ternary rings of projective planes. J. Algebra 319(2), 868–892 (2008)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bader L., Lunardon G.: On non-hyperelliptic flocks. Eur. J. Comb. 15(5), 411–415 (1994)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Baez J.C.: The octonions. Bull. Am. Math. Soc. (N.S.) 39(2), 145–205 (2002)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ball S., Ebert G., Lavrauw M.: A geometric construction of finite semifields. J. Algebra 311(1), 117–129 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bierbrauer J.: New semifields, PN and APN functions. Des. Codes Cryptogr. 54(3), 189–200 (2010)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bloemen I., Thas J.A., Van Maldeghem H.: Translation ovoids of generalized quadrangles and hexagons. Geom. Dedicata. 72(1), 19–62 (1998)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Blokhuis A., Lavrauw M., Ball S.: On the classification of semifield flocks. Adv. Math. 180(1), 104–111 (2003)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Bruck R.H., Bose R.C.: The construction of translation planes from projective spaces. J. Algebra 1, 85–102 (1964)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Bruck R.H., Kleinfeld E.: The structure of alternative division rings. Proc. Am. Math. Soc. 2, 878–890 (1951)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Budaghyan L., Helleseth T.: New commutative semifields defined by new PN multinomials. Cryptogr. Commun. 3(1), 1–16 (2011)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Cardinali I., Polverino O., Trombetti R.: Semifield planes of order q 4 with kernel \({F_{q^2}}\) and center F q. Eur. J. Comb. 27(6), 940–961 (2006)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Cohen S.D., Ganley M.J.: Commutative semifields, two-dimensional over their middle nuclei. J. Algebra 75(2), 373–385 (1982)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Coulter R.S., Henderson M.: Commutative presemifields and semifields. Adv. Math. 217(1), 282–304 (2008)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Cronheim A.: T-groups and their geometry. Ill. J. Math. 9, 1–30 (1965)MathSciNetMATHGoogle Scholar
  20. 20.
    Dembowski P.: Finite Geometries. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44. Springer, Berlin (1968)Google Scholar
  21. 21.
    Dempwolff U.: Semifield planes of order 81. J. Geom. 89(1–2), 1–16 (2008)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Dempwolff U.: On irreducible semilinear transformations. Forum Math. 22(6), 1193–1206 (2010)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Dickson L.E.: Linear algebras in which division is always uniquely possible. Trans. Am. Math. Soc. 7(3), 370–390 (1906)MATHCrossRefGoogle Scholar
  24. 24.
    Dickson L.E.: On commutative linear algebras in which division is always uniquely possible. Trans. Am. Math. Soc. 7(4), 514–522 (1906)MATHCrossRefGoogle Scholar
  25. 25.
    Ebert G., Marino G., Polverino O., Trombetti R.: On the multiplication of some semifields of order q 6. Finite Fields Appl. 15(2), 160–173 (2009)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Gow R., Sheekey J.: On primitive elements in finite semifields. Finite Fields Appl. 17(2), 194–204 (2011)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Hall M.: Projective planes. Trans. Am. Math. Soc. 54, 229–277 (1943)MATHCrossRefGoogle Scholar
  28. 28.
    Hiramine Y.: Automorphisms of $p$-groups of semifield type. Osaka J. Math. 20(4), 735–746 (1983)MathSciNetMATHGoogle Scholar
  29. 29.
    Hughes D.R., Kleinfeld E.: Seminuclear extensions of Galois fields. Am. J. Math. 82, 389–392 (1960)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Hughes D.R., Piper F.C.: Projective Planes. Graduate Texts in Mathematics, vol. 6. Springer, New York (1973).Google Scholar
  31. 31.
    Jha V., Johnson N.L.: The dimension of a subplane of a translation plane. Bull. Belgian Math. Soc. Simon Stevin 17(3), 463–477 (2010)MathSciNetMATHGoogle Scholar
  32. 32.
    Johnson N.L., Jha V., Biliotti M.: Handbook of Finite Translation Planes, Volume 289 of Pure and Applied Mathematics (Boca Raton). Chapman & Hall/CRC, Boca Raton (2007).Google Scholar
  33. 33.
    Johnson N.L., Marino G., Polverino O., Trombetti R.: Semifields of order q 6 with left nucleus \({\mathbb{F}_{q^3}}\) and center \({\mathbb{F}_q}\). Finite Fields Appl. 14(2), 456–469 (2008)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Johnson N.L., Marino G., Polverino O., Trombetti R.: On a generalization of cyclic semifields. J. Algebraic Comb. 29(1), 1–34 (2009)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Kantor W.M.: Commutative semifields and symplectic spreads. J. Algebra 270(1), 96–114 (2003)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Kantor W.M.: Finite semifields. In: Hulpke, A., Liebler, R., Penttila, T., Serres, À. (eds.) Finite Geometries, Groups, and Computation, pp. 103–114. Walter de Gruyter GmbH & Co. KG, Berlin (2006).Google Scholar
  37. 37.
    Kantor W.M.: HMO-planes. Adv. Geom. 9(1), 31–43 (2009)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Kantor W.M., Liebler R.A.: Semifields arising from irreducible semilinear transformations. J. Aust. Math. Soc. 85(3), 333–339 (2008)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Kaplansky I.: Infinite-dimensional quadratic forms admitting composition. Proc. Am. Math. Soc. 4, 956–960 (1953)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Kleinfeld E.: Techniques for enumerating Veblen-Wedderburn systems. J. Assoc. Comput. Mach. 7, 330–337 (1960)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Knuth D.E.: Finite semifields and projective planes—PhD, pp. 1–70. PhD dissertation (1963).Google Scholar
  42. 42.
    Knuth D.E.: Finite semifields and projective planes. J. Algebra 2, 182–217 (1965)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Lavrauw M.: Scattered spaces with respect to spreads, and eggs in finite projective spaces. Dissertation, Eindhoven University of Technology, Eindhoven (2001).Google Scholar
  44. 44.
    Lavrauw M.: The two sets of three semifields associated with a semifield flock. Innov. Incidence Geom. 2, 101–107 (2005)MathSciNetMATHGoogle Scholar
  45. 45.
    Lavrauw M.: Sublines of prime order contained in the set of internal points of a conic. Des. Codes Cryptogr. 38(1), 113–123 (2006)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Lavrauw M.: On the isotopism classes of finite semifields. Finite Fields Appl. 14(4), 897–910 (2008)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Lavrauw M.: Finite semifields with a large nucleus and higher secant varieties to Segre varieties. Adv. Geom. 11, 399–410 (2011)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Lavrauw M., Polverino O.: Finite semifields and Galois geometry. In: De Beule J., Storme L. (eds.) Current Research Topics in Galois Geometry. NOVA Academic Publishers. ISBN 978-1-61209-523-3 (2011).Google Scholar
  49. 49.
    Lavrauw M., Sheekey J.: Semifields from skew polynomial rings. Adv. Geom. (to appear).Google Scholar
  50. 50.
    Lavrauw M., Vande Voorde G.: On linear sets on a projective line. Des. Codes Cryptogr. 56(2–3), 89–104 (2010)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Liebler R.A.: Autotopism group representations. J. London Math. Soc. (2) 23(1), 85–91 (1981)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Liebler R.A.: On nonsingular tensors and related projective planes. Geom. Dedicata 11(4), 455–464 (1981)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Lunardon G., Marino G., Polverino O., Trombetti R.: Translation dual of a semifield. J. Comb. Theory Ser. A. 115(8), 1321–1332 (2008)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Lunardon G., Marino G., Polverino O., Trombetti R.: Symplectic semifield spreads of PG(5, q) and the Veronese surface. Ric. Mat. 60(1), 125–142 (2011)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Maduram D.M.: Transposed translation planes. Proc. Am. Math. Soc. 53(2), 265–270 (1975)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Marino G., Polverino O.: On the nuclei of a finite semifield. Preprint.Google Scholar
  57. 57.
    Marino G., Polverino O., Trombetti R.: On \({\mathbb{F}_q}\) -linear sets of PG(3, q 3) and semifields. J. Comb. Theory Ser. A. 114(5), 769–788 (2007)MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Menichetti G.: On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field. J. Algebra 47(2), 400–410 (1977)MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Menichetti G.: n-Dimensional algebras over a field with a cyclic extension of degree n. Geom. Dedicata 63(1), 69–94 (1996)MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Nagy G.P.: On the multiplication groups of semifields. Eur. J. Comb. 31(1), 18–24 (2010)MATHCrossRefGoogle Scholar
  61. 61.
    Penttila T., Williams B.: Ovoids of parabolic spaces. Geom. Dedicata 82(1–3), 1–19 (2000)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Polverino O.: Linear sets in finite projective spaces. Discret. Math. 310(22), 3096–3107 (2010)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Polverino O., Trombetti R.: Fractional dimension of binary Knuth semifield planes. J. Comb. Des. 20(7), 317–327 (2012)MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Pott A., Zhou Y.: A character theoretic approach to planar functions. Cryptogr. Commun. 3, 293–300 (2011)MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Rúa I.R.: Primitive and non primitive finite semifields. Commun. Algebra 32(2), 793–803 (2004)MATHCrossRefGoogle Scholar
  66. 66.
    Rúa I.F., Combarro E.F., Ranilla J.: Classification of semifields of order 64. J. Algebra 322(11), 4011–4029 (2009)MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Rúa I.F., Combarro E.F., Ranilla J.: Computational methods for finite semifields. In: Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering, pp. 937–1461. CMMCSE (2009).Google Scholar
  68. 68.
    Skornyakov L.A.: Alternative fields. Ukrain. Mat. Žurnal. 2, 70–85 (1950)MathSciNetMATHGoogle Scholar
  69. 69.
    Thas J.A.: Generalized quadrangles of order (s, s 2). I. J. Comb. Theory Ser. A. 67(2), 140–160 (1994)MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    Thas J.A.: Generalized quadrangles of order (s, s 2). II. J. Comb. Theory Ser. A. 79(2), 223–254 (1997)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    Walker R.J.: Determination of division algebras with 32 elements. In: Proceedings of Symposia in Applied Mathematics, vol. XV, pp. 83–85. American Mathematical Society, Providence (1963).Google Scholar
  72. 72.
    Zorn M.: Theorie der alternativen ringe. Abhandlungen aus dem Mathematischen Seminar der Universitt Hamburg. 8, 123–147 (1931)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Università degli Studi di PadovaPaduaItaly

Personalised recommendations