Designs, Codes and Cryptography

, Volume 67, Issue 3, pp 375–384 | Cite as

Families of twisted tensor product codes

  • Luca Giuzzi
  • Valentina PepeEmail author


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.


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

Mathematics Subject Classification (2010)

94B05 94B27 15A69 51E20 


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  1. 1.
    Assmus E.F., Key J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  2. 2.
    Betten A.: Twisted tensor product codes. Des. Codes Cryptogr. 47(1–3), 191–219 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Berlekamp E.R.: Algebraic Coding Theory. McGraw-Hill, New York (1968)zbMATHGoogle Scholar
  4. 4.
    Bose R.C.: An affine analogue of Singer’s theorem. J. Indian Math. Soc. 6, 1–15 (1942)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cossidente A., Labbate D., Siciliano A.: Veronese varieties over finite fields and their projections. Des. Codes Cryptogr. 22, 19–32 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Couvreur A., Duursma I.: Evaluation codes from smooth Quadric surfaces and twisted segre varieties, arXiv 1101.4603v1.Google Scholar
  7. 7.
    Giuzzi L., Sonnino A.: LDPC codes from singer cycles. Discret. Appl. Math. 157(8), 1723–1728 (2009)MathSciNetzbMATHCrossRefGoogle 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)zbMATHCrossRefGoogle 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)zbMATHGoogle Scholar
  12. 12.
    Kantor W.M., Shult E.E.: Veroneseans, power subspaces and independence. Adv. Geom. (in press).Google Scholar
  13. 13.
    Lunardon G.: Planar fibrations and algebraic subvarieties of the Grassmann variety. Geom. Dedicata 16(3), 291–313 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lunardon G.: Normal spreads. Geom. Dedicata 75(3), 245–261 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error Correcting Codes. North-Holland, Amsterdam (1977)zbMATHGoogle Scholar
  16. 16.
    Paige L.J.: A note on the Mathieu groups. Can. J. Math. 9, 15–18 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Pepe V.: On the algebraic variety \({{\mathcal V}_{r,t}}\) . Finite Fields Appl. 17(4), 343–349 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Radkova D., Van Zanten A.J.: Constacyclic codes as invariant subspaces. Linear Algebra Appl. 430(2–3), 855–864 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Segre B.: Teoria di Galois, fibrazioni proiettive e geometrie non Desarguesiane. Ann. Mat. Pura Appl. 64, 1–76 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Snapper E.: Periodic linear transformations of affine and projective geometries. Can. J. Math. 2, 149–151 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Steinberg R.: Representations of algebraic groups. Nagoya Math. J. 22, 33–56 (1963)MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Facoltà di IngegneriaUniversità degli Studi di BresciaBresciaItaly
  2. 2.Department of MathematicsUniversiteit GentGentBelgium

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