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Non-free extensions of the simplex codes over a chain ring with four elements

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Abstract

Let R be a chain ring with four elements. In this paper, we present two new constructions of R-linear codes that contain a subcode associated with a simplex code over the ring R. The simplex codes are defined as the codes generated by a matrix having as columns the homogeneous coordinates of all points in some projective Hjelmslev geometry PHG(R k). The first construction generalizes a recent result by Kiermaier and Zwanzger to codes of arbitrary dimension. We provide a geometric interpretation of their construction which is then extended to projective Hjelmslev spaces of arbitrary dimension. The second construction exploits the possibility of adding two non-free rows to the generator matrix of a linear code over R associated with a given point set. Though the construction works over both chain rings with four elements, the better codes are obtained for \({R=\mathbb{Z}_4}\) .

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References

  1. Artmann B.: Hjelmslev-Ebenen mit verfeinerten Nachbarschaftsrelationen. Math. Z. 112, 163–180 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonnecaze A., Duursma I.M.: Translates of linear codes over \({\mathbb {Z}_4}\) . IEEE Trans. Inf. Theory 43(4), 1218–1230 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cronheim A.: Dual numbers, Witt vectors, and Hjelmslev planes. Geom. Dedicata 7, 287–302 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Drake D.A.: On n-uniform Hjelmslev planes. J. Comb. Theory 9, 267–288 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grassl M.: Tables of bounds on linear and quantum error-correcting codes, http://www.codetables.de/ (2008).

  6. Honold T., Kiermaier M.: Classification of maximal arcs in small projective Hjelmslev geometries. In: Proceedings of the Tenth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 112–117. Zvenigorod, Russia (2006).

  7. Honold T., Kiermaier M.: The existence of maximal (q 2,2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic. Des. Codes Cryptogr. doi:10.1007/s10623-012-9653-y.

  8. Honold T., Landjev I.: Linearly representable codes over chain rings. Abhandlungen aus dem mathematischen Seminar der Universität Hamburg 69, 187–203 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Honold T., Landjev I. Linear codes over finite chain rings. Electron. J. Comb. 7(research paper 11), p. 22 (electronic) (2000).

  10. Honold T., Landjev I.: On arcs in projective Hjelmslev planes. Discret. Math. 231(1–3), 265–278 (2001). 17th British Combinatorial Conference, University of Kent, Canterbury (1999).

    Google Scholar 

  11. Honold T., Landjev I.: On maximal arcs in projective Hjelmslev planes over chain rings of even characteristic. Finite Fields Appl. 11(2), 292–304 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Honold T., Landjev I.: Linear codes over finite chain rings and projective Hjelmslev geometries. In: ~Solé P. (ed.), Codes Over Rings, vol.~6 of Series on Coding Theory and Cryptology. World Scientific, Singapore (2009).

  13. Honold T., Landjev I.: Codes over rings and ring geometries. In: Beule J.D., Storme L. (eds.) Current Research Topics in Galois Geometry, chap.~7, pp. 159–184. Nova Science Publishers, New York (2011).

  14. Kiermaier M., Kohnert A.: On-line tables of arcs in projective Hjelmslev planes, http://www.algorithm.uni-bayreuth.de/en/research/Coding_Theory/PHG_arc_table/ (2011).

  15. Kiermaier M., Zwanzger J.: A non-free \({\mathbb{Z}_4}\) -linear code of high minimum Lee distance. Adv. Math. Commun. 5(2), 275–286 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kreuzer A.: Hjelmslev-Räume. Result. der Math. 12, 148–156 (1987)

    MathSciNet  MATH  Google Scholar 

  17. Kreuzer A.: Projektive Hjelmslev-Räume. Dissertation, Technische Universität München (1988).

  18. Kreuzer A.: Fundamental theorem of projective Hjelmslev spaces. Mitteilungen der Math. Ges. Hambg. 12(3), 809–817 (1991)

    MathSciNet  MATH  Google Scholar 

  19. Kreuzer A.: A system of axioms for projective Hjelmslev spaces. J. Geom 40, 125–147 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Landjev I., Honold T.: Arcs in projective Hjelmslev planes. Discret. Math. Appl. 11(1), 53–70 (2001). Originally published in Diskretnaya Matematika 131, 90–109 (2001) (in Russian).

    Google Scholar 

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Authors and Affiliations

  1. Department of Information Science and Electronics Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, 310027, China

    Thomas Honold

  2. New Bulgarian University, 21 Montevideo Str., 1618, Sofia, Bulgaria

    Ivan Landjev

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  1. Thomas Honold
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  2. Ivan Landjev
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Correspondence to Ivan Landjev.

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

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Honold, T., Landjev, I. Non-free extensions of the simplex codes over a chain ring with four elements. Des. Codes Cryptogr. 66, 27–38 (2013). https://doi.org/10.1007/s10623-012-9649-7

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  • DOI: https://doi.org/10.1007/s10623-012-9649-7

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