Families of twisted tensor product codes

Abstract

Using geometric properties of the variety \({\mathcal V_{r,t}}\) , the image under the Grassmannian map of a Desarguesian (t − 1)-spread of PG(rt − 1, q), we introduce error correcting codes related to the twisted tensor product construction, producing several families of constacyclic codes. We determine the precise parameters of these codes and characterise the words of minimum weight.

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References

  1. 1.

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

    Google Scholar 

  2. 2.

    Betten A.: Twisted tensor product codes. Des. Codes Cryptogr. 47(1–3), 191–219 (2008)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Berlekamp E.R.: Algebraic Coding Theory. McGraw-Hill, New York (1968)

    Google Scholar 

  4. 4.

    Bose R.C.: An affine analogue of Singer’s theorem. J. Indian Math. Soc. 6, 1–15 (1942)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Cossidente A., Labbate D., Siciliano A.: Veronese varieties over finite fields and their projections. Des. Codes Cryptogr. 22, 19–32 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Couvreur A., Duursma I.: Evaluation codes from smooth Quadric surfaces and twisted segre varieties, arXiv 1101.4603v1.

  7. 7.

    Giuzzi L., Sonnino A.: LDPC codes from singer cycles. Discret. Appl. Math. 157(8), 1723–1728 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Harris J.: Algebraic Geometry: a First course. GTM 133. Springer, New York (1992)

    Google Scholar 

  9. 9.

    Hassett B.: Introduction to Algebraic Geometry. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  10. 10.

    Hirschfeld J.W.P.: Finite Projective Spaces of Three Dimension. Oxford University Press, Oxford (1986)

    Google Scholar 

  11. 11.

    Hirschfeld J.W.P., Thas J.A.: General Galois Geometries. Oxford University Press, Oxford (1991)

    Google Scholar 

  12. 12.

    Kantor W.M., Shult E.E.: Veroneseans, power subspaces and independence. Adv. Geom. (in press).

  13. 13.

    Lunardon G.: Planar fibrations and algebraic subvarieties of the Grassmann variety. Geom. Dedicata 16(3), 291–313 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Lunardon G.: Normal spreads. Geom. Dedicata 75(3), 245–261 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

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

    Google Scholar 

  16. 16.

    Paige L.J.: A note on the Mathieu groups. Can. J. Math. 9, 15–18 (1957)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Pepe V.: On the algebraic variety \({{\mathcal V}_{r,t}}\) . Finite Fields Appl. 17(4), 343–349 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Radkova D., Van Zanten A.J.: Constacyclic codes as invariant subspaces. Linear Algebra Appl. 430(2–3), 855–864 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Segre B.: Teoria di Galois, fibrazioni proiettive e geometrie non Desarguesiane. Ann. Mat. Pura Appl. 64, 1–76 (1964)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Snapper E.: Periodic linear transformations of affine and projective geometries. Can. J. Math. 2, 149–151 (1950)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Steinberg R.: Representations of algebraic groups. Nagoya Math. J. 22, 33–56 (1963)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Valentina Pepe.

Additional information

L. Giuzzi—Part of this research has been performed while a guest of the Department of Mathematics of Ghent University.

Communicated by G. Lunardon.

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Giuzzi, L., Pepe, V. Families of twisted tensor product codes. Des. Codes Cryptogr. 67, 375–384 (2013). https://doi.org/10.1007/s10623-012-9613-6

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Keywords

  • Segre product
  • Veronesean
  • Grassmannian
  • Desarguesian spread
  • Subgeometry
  • Twisted product
  • Constacyclic error correcting code
  • Minimum weight

Mathematics Subject Classification (2010)

  • 94B05
  • 94B27
  • 15A69
  • 51E20