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The structure of dual Grassmann codes

Abstract

In this article we study the duals of Grassmann codes, certain codes coming from the Grassmannian variety. Exploiting their structure, we are able to count and classify all their minimum weight codewords. In this classification the lines lying on the Grassmannian variety play a central role. Related codes, namely the affine Grassmann codes, were introduced more recently in Beelen et al. (IEEE Trans Inf Theory 56(7):3166–3176, 2010), while their duals were introduced and studied in Beelen et al. (IEEE Trans Inf Theory 58(6):3843–3855, 2010). In this paper we also classify and count the minimum weight codewords of the dual affine Grassmann codes. Combining the above classification results, we are able to show that the dual of a Grassmann code is generated by its minimum weight codewords. We use these properties to establish that the increase of value of successive generalized Hamming weights of a dual Grassmann code is 1 or 2.

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References

  1. 1.

    Beelen P., Ghorpade S.R., Høholdt T.: Affine Grassmann codes. IEEE Trans. Inf. Theory, 56(7), 3166–3176 (2010).

  2. 2.

    Beelen P., Ghorpade S.R., Høholdt T.: Duals of affine Grassmann codes and their relatives. IEEE Trans. Inf. Theory, 58(6), 3843–3855 (2012).

  3. 3.

    Ghorpade S.R., Lachaud G.: Higher weights of Grassmann codes. In: Coding Theory, Cryptography and Related Areas (Guanajuato, 1998), pp. 122–131. Springer, Berlin (2000).

  4. 4.

    Ghorpade S.R., Kaipa K.V.: Automorphism groups of Grassmann codes. Finite Fields Appl. 23, 80–102 (2013).

  5. 5.

    Ghorpade S.R., Patil A.R., Pillai H.K.: Decomposable subspaces, linear sections of Grassmann varieties, and higher weights of Grassmann codes. Finite Fields Appl. 15, 54–68 (2009).

  6. 6.

    Hansen J.P., Johnsen T., Ranestad K.: Grassmann codes and Schubert unions. In: Arithmetic, Geometry and Coding Theory (Luminy, 2005), Séminaires et Congrès, vol. 21, pp. 103–121. Société Mathématique de France, Paris (2009).

  7. 7.

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

  8. 8.

    Nogin D.Y.: Codes associated to Grassmannians. In: Arithmetic, Geometry and Coding Theory (Luminy, 1993), pp. 145–154. Walter de Gruyter, Berlin (1996).

  9. 9.

    Pankov M.: Grassmannians of Classical Buildings. World Scientific, Singapore (2010).

  10. 10.

    Ryan C.T.: An application of Grassmannian varieties to coding theory. Congr. Numer. 57, 257–271 (1987).

  11. 11.

    Ryan C.T.: Projective codes based on Grassmannian varieties. Congr. Numer. 57, 273–279 (1987).

  12. 12.

    Ryan C.T., Ryan K.M.: The minimum weight of the Grassmann codes \(C(k, n)\). Discret. Appl. Math. 28(2), 149–156 (1990).

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Acknowledgments

We would like to thank professor Sudhir R. Ghorpade for pleasant discussions on several topics related to Grassmann codes. The authors gratefully acknowledge the support from the Danish National Research Foundation and the National Science Foundation of China (Grant No. 11061130539) for the Danish-Chinese Center for Applications of Algebraic Geometry in Coding Theory and Cryptography as well as the support from The Danish Council for Independent Research (Grant No. DFF–4002-00367).

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Correspondence to Peter Beelen.

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

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Beelen, P., Piñero, F. The structure of dual Grassmann codes. Des. Codes Cryptogr. 79, 451–470 (2016). https://doi.org/10.1007/s10623-015-0085-3

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Keywords

  • Dual Grassmann code
  • Hamming weights
  • Tanner code

Mathematics Subject Classification

  • 14G50
  • 94B27
  • 14M15