Skip to main content
Log in

Finite semifields and nonsingular tensors

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Albert A.A.: Non-associative algebras. I. Fundamental concepts and isotopy. Ann. Math. (2) 43, 685–707 (1942)

    Article  MATH  Google Scholar 

  2. Albert A.A.: On nonassociative division algebras. Trans. Am. Math. Soc. 72, 296–309 (1952)

    Article  MATH  Google Scholar 

  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).

  4. Albert A.A.: Generalized twisted fields. Pac. J. Math. 11, 1–8 (1961)

    Article  MATH  Google Scholar 

  5. André J.: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Math. Z. 60, 156–186 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aschbacher M.: Isotopy and geotopy for ternary rings of projective planes. J. Algebra 319(2), 868–892 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bader L., Lunardon G.: On non-hyperelliptic flocks. Eur. J. Comb. 15(5), 411–415 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Baez J.C.: The octonions. Bull. Am. Math. Soc. (N.S.) 39(2), 145–205 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ball S., Ebert G., Lavrauw M.: A geometric construction of finite semifields. J. Algebra 311(1), 117–129 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bierbrauer J.: New semifields, PN and APN functions. Des. Codes Cryptogr. 54(3), 189–200 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bloemen I., Thas J.A., Van Maldeghem H.: Translation ovoids of generalized quadrangles and hexagons. Geom. Dedicata. 72(1), 19–62 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Blokhuis A., Lavrauw M., Ball S.: On the classification of semifield flocks. Adv. Math. 180(1), 104–111 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bruck R.H., Bose R.C.: The construction of translation planes from projective spaces. J. Algebra 1, 85–102 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bruck R.H., Kleinfeld E.: The structure of alternative division rings. Proc. Am. Math. Soc. 2, 878–890 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  15. Budaghyan L., Helleseth T.: New commutative semifields defined by new PN multinomials. Cryptogr. Commun. 3(1), 1–16 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cohen S.D., Ganley M.J.: Commutative semifields, two-dimensional over their middle nuclei. J. Algebra 75(2), 373–385 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Coulter R.S., Henderson M.: Commutative presemifields and semifields. Adv. Math. 217(1), 282–304 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cronheim A.: T-groups and their geometry. Ill. J. Math. 9, 1–30 (1965)

    MathSciNet  MATH  Google Scholar 

  20. Dembowski P.: Finite Geometries. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44. Springer, Berlin (1968)

    Google Scholar 

  21. Dempwolff U.: Semifield planes of order 81. J. Geom. 89(1–2), 1–16 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dempwolff U.: On irreducible semilinear transformations. Forum Math. 22(6), 1193–1206 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dickson L.E.: Linear algebras in which division is always uniquely possible. Trans. Am. Math. Soc. 7(3), 370–390 (1906)

    Article  MATH  Google Scholar 

  24. Dickson L.E.: On commutative linear algebras in which division is always uniquely possible. Trans. Am. Math. Soc. 7(4), 514–522 (1906)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gow R., Sheekey J.: On primitive elements in finite semifields. Finite Fields Appl. 17(2), 194–204 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hall M.: Projective planes. Trans. Am. Math. Soc. 54, 229–277 (1943)

    Article  MATH  Google Scholar 

  28. Hiramine Y.: Automorphisms of $p$-groups of semifield type. Osaka J. Math. 20(4), 735–746 (1983)

    MathSciNet  MATH  Google Scholar 

  29. Hughes D.R., Kleinfeld E.: Seminuclear extensions of Galois fields. Am. J. Math. 82, 389–392 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hughes D.R., Piper F.C.: Projective Planes. Graduate Texts in Mathematics, vol. 6. Springer, New York (1973).

  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)

    MathSciNet  MATH  Google Scholar 

  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).

  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)

    Article  MathSciNet  MATH  Google Scholar 

  34. Johnson N.L., Marino G., Polverino O., Trombetti R.: On a generalization of cyclic semifields. J. Algebraic Comb. 29(1), 1–34 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kantor W.M.: Commutative semifields and symplectic spreads. J. Algebra 270(1), 96–114 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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).

  37. Kantor W.M.: HMO-planes. Adv. Geom. 9(1), 31–43 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kantor W.M., Liebler R.A.: Semifields arising from irreducible semilinear transformations. J. Aust. Math. Soc. 85(3), 333–339 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kaplansky I.: Infinite-dimensional quadratic forms admitting composition. Proc. Am. Math. Soc. 4, 956–960 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kleinfeld E.: Techniques for enumerating Veblen-Wedderburn systems. J. Assoc. Comput. Mach. 7, 330–337 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  41. Knuth D.E.: Finite semifields and projective planes—PhD, pp. 1–70. PhD dissertation (1963).

  42. Knuth D.E.: Finite semifields and projective planes. J. Algebra 2, 182–217 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lavrauw M.: Scattered spaces with respect to spreads, and eggs in finite projective spaces. Dissertation, Eindhoven University of Technology, Eindhoven (2001).

  44. Lavrauw M.: The two sets of three semifields associated with a semifield flock. Innov. Incidence Geom. 2, 101–107 (2005)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  46. Lavrauw M.: On the isotopism classes of finite semifields. Finite Fields Appl. 14(4), 897–910 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. Lavrauw M.: Finite semifields with a large nucleus and higher secant varieties to Segre varieties. Adv. Geom. 11, 399–410 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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).

  49. Lavrauw M., Sheekey J.: Semifields from skew polynomial rings. Adv. Geom. (to appear).

  50. Lavrauw M., Vande Voorde G.: On linear sets on a projective line. Des. Codes Cryptogr. 56(2–3), 89–104 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  51. Liebler R.A.: Autotopism group representations. J. London Math. Soc. (2) 23(1), 85–91 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  52. Liebler R.A.: On nonsingular tensors and related projective planes. Geom. Dedicata 11(4), 455–464 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  53. Lunardon G., Marino G., Polverino O., Trombetti R.: Translation dual of a semifield. J. Comb. Theory Ser. A. 115(8), 1321–1332 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  55. Maduram D.M.: Transposed translation planes. Proc. Am. Math. Soc. 53(2), 265–270 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  56. Marino G., Polverino O.: On the nuclei of a finite semifield. Preprint.

  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)

    Article  MathSciNet  MATH  Google Scholar 

  58. Menichetti G.: On a Kaplansky conjecture concerning three-dimensional division algebras over a finite field. J. Algebra 47(2), 400–410 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  59. Menichetti G.: n-Dimensional algebras over a field with a cyclic extension of degree n. Geom. Dedicata 63(1), 69–94 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  60. Nagy G.P.: On the multiplication groups of semifields. Eur. J. Comb. 31(1), 18–24 (2010)

    Article  MATH  Google Scholar 

  61. Penttila T., Williams B.: Ovoids of parabolic spaces. Geom. Dedicata 82(1–3), 1–19 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  62. Polverino O.: Linear sets in finite projective spaces. Discret. Math. 310(22), 3096–3107 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  63. Polverino O., Trombetti R.: Fractional dimension of binary Knuth semifield planes. J. Comb. Des. 20(7), 317–327 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  64. Pott A., Zhou Y.: A character theoretic approach to planar functions. Cryptogr. Commun. 3, 293–300 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  65. Rúa I.R.: Primitive and non primitive finite semifields. Commun. Algebra 32(2), 793–803 (2004)

    Article  MATH  Google Scholar 

  66. Rúa I.F., Combarro E.F., Ranilla J.: Classification of semifields of order 64. J. Algebra 322(11), 4011–4029 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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).

  68. Skornyakov L.A.: Alternative fields. Ukrain. Mat. Žurnal. 2, 70–85 (1950)

    MathSciNet  MATH  Google Scholar 

  69. Thas J.A.: Generalized quadrangles of order (s, s 2). I. J. Comb. Theory Ser. A. 67(2), 140–160 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  70. Thas J.A.: Generalized quadrangles of order (s, s 2). II. J. Comb. Theory Ser. A. 79(2), 223–254 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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).

  72. Zorn M.: Theorie der alternativen ringe. Abhandlungen aus dem Mathematischen Seminar der Universitt Hamburg. 8, 123–147 (1931)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michel Lavrauw.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lavrauw, M. Finite semifields and nonsingular tensors. Des. Codes Cryptogr. 68, 205–227 (2013). https://doi.org/10.1007/s10623-012-9710-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-012-9710-6

Keywords

Mathematics Subject Classification

Navigation