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A geometric approach to rank metric codes and a classification of constant weight codes

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Abstract

In this work we develop a geometric approach to the study of rank metric codes. Using this method, we introduce a simpler definition for generalized rank weight of linear codes. We give a complete classification of constant rank weight code and we give their generalized rank weights.

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References

  1. Bonisoli A.: Every equidistant linear code is a sequence of dual Hamming codes. ARS Comb. 18, 181–186 (1983).

    MathSciNet  MATH  Google Scholar 

  2. Cai N., Yeung R.W.: Secure network coding. In: Proceedings IEEE International Symposium on Information Theory, p. 323 (2002). https://doi.org/10.1109/ISIT.2002.1023595.

  3. Csajbók B., Marino G., Polverino O., Zullo F.: Maximum scattered linear sets and MRD-codes. J. Algebr. Comb. 46(3), 517–531 (2017). https://doi.org/10.1007/s10801-017-0762-6.

    Article  MathSciNet  MATH  Google Scholar 

  4. Csajbók B., Marino G., Polverino O., Zullo F.: A special class of scattered subspaces (2019).

  5. Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25(3), 226–241 (1978). https://doi.org/10.1016/0097-3165(78)90015-8.

    Article  MathSciNet  MATH  Google Scholar 

  6. Ducoat J., Kyureghyan G.: Generalized rank weights: a duality statement. Top. Finite Fields 632, 101–109 (2015).

    MathSciNet  MATH  Google Scholar 

  7. El Rouayheb S.Y., Soljanin E.: On wiretap networks II. In: 2007 IEEE International Symposium on Information Theory, pp. 551–555 (2007).https://doi.org/10.1109/ISIT.2007.4557098.

  8. Gabidulin E.: Theory of codes with maximum rank distance (translation). Problems Inform. Transm. 21, 1–12 (1985).

    MathSciNet  MATH  Google Scholar 

  9. Gabidulin E.M., Paramonov A.V., Tretjakov O.V.: Ideals over a non-commutative ring and their application in cryptology. In: Davies D.W. (ed.) Advances in Cryptology—EUROCRYPT ’91, pp. 482–489. Springer, Berlin (1991).

    Chapter  Google Scholar 

  10. Giuzzi L., Zullo F.: Identifiers for MRD-codes. Linear Algebra Appl. 575, 66–86 (2019). https://doi.org/10.1016/j.laa.2019.03.030.

    Article  MathSciNet  MATH  Google Scholar 

  11. Helleseth T., Kløve T., Mykkeltveit J.: The weight distribution of irreducible cyclic codes with block lengths \(n_1((q^1-1)N)\). Discret. Math. 18(2), 179–211 (1977). https://doi.org/10.1016/0012-365X(77)90078-4.

    Article  MATH  Google Scholar 

  12. Hill R.: Caps and codes. Discret. Math. 22(2), 111–137 (1978). https://doi.org/10.1016/0012-365X(78)90120-6.

    Article  MathSciNet  MATH  Google Scholar 

  13. Jurrius R., Pellikaan R.: The extended and generalized rank weight enumerator of a code. ACM Commun. Comput. Algebra 49(1), 21–21 (2015). https://doi.org/10.1145/2768577.2768605.

    Article  MATH  Google Scholar 

  14. Jurrius R., Pellikaan R.: On defining generalized rank weights. Adv. Math. Commun. 11, 225 (2017). https://doi.org/10.3934/amc.2017014.

    Article  MathSciNet  MATH  Google Scholar 

  15. Kurihara J., Matsumoto R., Uyematsu T.: Relative generalized rank weight of linear codes and its applications to network coding. IEEE Trans. Inform. Theory 61(7), 3912–3936 (2015). https://doi.org/10.1109/TIT.2015.2429713.

    Article  MathSciNet  MATH  Google Scholar 

  16. Lavrauw M., Van de Voorde G.: Field reduction and linear sets in finite geometry. Top. Finite Fields 632, 271–293 (2015).

    MathSciNet  MATH  Google Scholar 

  17. Liu Z., Chen W.: Notes on the value function. Des. Codes Cryptogr. 54(1), 11 (2009). https://doi.org/10.1007/s10623-009-9305-z.

    Article  MathSciNet  MATH  Google Scholar 

  18. Lunardon G.: Normal spreads. Geom. Dedicata 75(3), 245–261 (1999). https://doi.org/10.1023/A:1005052007006.

    Article  MathSciNet  MATH  Google Scholar 

  19. Lunardon G.: MRD-codes and linear sets. J. Comb. Theory Ser. A 149, 1–20 (2017). https://doi.org/10.1016/j.jcta.2017.01.002.

    Article  MathSciNet  MATH  Google Scholar 

  20. Martínez-Peñas U.: On the similarities between generalized rank and Hamming weights and their applications to network coding. IEEE Trans. Inform. Theory 62(7), 4081–4095 (2016). https://doi.org/10.1109/TIT.2016.2570238.

    Article  MathSciNet  MATH  Google Scholar 

  21. Morrison K.: Equivalence for rank-metric and matrix codes and automorphism groups of Gabidulin codes. IEEE Trans. Inform. Theory 60(11), 7035–7046 (2014). https://doi.org/10.1109/TIT.2014.2359198.

    Article  MathSciNet  MATH  Google Scholar 

  22. Oggier F., Sboui A.: On the existence of generalized rank weights. In: 2012 International Symposium on Information Theory and its Applications, pp. 406–410 (2012).

  23. Ozarow L.H., Wyner A.D.: Wire-tap Channel II. In: Beth T., Cot N., Ingemarsson I. (eds.) Advances in Cryptology, pp. 33–50. Springer, Berlin (1985).

    Chapter  Google Scholar 

  24. Polverino O.: Linear sets in finite projective spaces. Discret. Math. 310(22), 3096–3107 (2010). https://doi.org/10.1016/j.disc.2009.04.007. (Combinatorics 2008).

    Article  MathSciNet  MATH  Google Scholar 

  25. Ravagnani A.: Generalized weights: an anticode approach. J. Pure Appl. Algebra 220(5), 1946–1962 (2016). https://doi.org/10.1016/j.jpaa.2015.10.009.

    Article  MathSciNet  MATH  Google Scholar 

  26. Sheekey J.: A new family of linear maximum rank distance codes. Adv. Math. Commun. 10, 475 (2016). https://doi.org/10.3934/amc.2016019.

    Article  MathSciNet  MATH  Google Scholar 

  27. Sheekey J.: Mrd codes: constructions and connections. In: Schmidt K.U., Winterhof A. (eds.) Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications, vol. 23, pp. 255–286. De Gruyter, Berlin (2019). (chap. 13).

    Chapter  Google Scholar 

  28. Sheekey J.: New semifields and new MRD codes from skew polynomial rings. J. Lond. Math. Soc. (2019). https://doi.org/10.1112/jlms.12281.

    Article  MATH  Google Scholar 

  29. Sheekey J., Van de Voorde G.: Rank-metric codes, linear sets, and their duality. Des. Codes Cryptogr. (2019). https://doi.org/10.1007/s10623-019-00703-z.

  30. Tsfasman M.A., VlăduŢ S.G.: Codes and Their Parameters, pp. 5–35. Springer, Dordrecht (1991). https://doi.org/10.1007/978-94-011-3810-9_1.

    Book  Google Scholar 

  31. Tsfasman M.A., Vladut S.G.: Geometric approach to higher weights. IEEE Trans. Inform. Theory 41(6), 1564–1588 (1995).

    Article  MathSciNet  Google Scholar 

  32. Wei V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inform. Theory 37(5), 1412–1418 (1991). https://doi.org/10.1109/18.133259.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

I would like to thank Rakhi Pratihar and Prof. Sudhir Ghorparde for their valuable comments and suggestions on this work. I also would like to thank the anonymous reviewer who introduced me to linear sets.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India

    Tovohery Hajatiana Randrianarisoa

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  1. Tovohery Hajatiana Randrianarisoa
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Correspondence to Tovohery Hajatiana Randrianarisoa.

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Communicated by I. Landjev.

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The author is supported by the Swiss National Science Foundation Grant No. 181446.

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Randrianarisoa, T.H. A geometric approach to rank metric codes and a classification of constant weight codes. Des. Codes Cryptogr. 88, 1331–1348 (2020). https://doi.org/10.1007/s10623-020-00750-x

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