Skip to main content
SpringerLink
Log in

Codes with the rank metric and matroids

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

We study the relationship between linear codes with the rank metric in the vector space of matrices with entries in a finite field and a q-analogue of matroids. We prove a Greene type identity for the rank generating function of these matroidal structures and the rank weight enumerator of these linear codes. As an application, we give a combinatorial proof of the MacWilliams type identity for Delsarte rank-metric codes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aigner M.: Combinatorial Theory, Grundlehren der Math. Wiss, vol. 234. Springer, Berlin (1979).

    Google Scholar 

  2. Barg A.: The matroid of supports of a linear code. Appl. Algebra Engrg. Commun. Comput. 8, 165–172 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  3. Britz T.: Code enumerators and Tutte polynomials. IEEE Trans. Inform. Theory 56, 4350–4358 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  4. Britz T., Shiromoto K.: On the covering dimension of a linear code. IEEE Trans. Inform. Theory 62, 2694–2701 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  5. Britz T., Shiromoto K., Westerbäck T.: Demi-matroids from codes over finite Frobenius rings. Des. Codes Cryptogr. 75, 97–107 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  6. Chan T.H., Grant A., Britz T.: Quasi-uniform codes and their applications. IEEE Trans. Inform. Theory 59, 7915–7926 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  7. Cruz J., Gorla E., López H.H., Ravagnani A.: Weight distribution of rank-metric codes. Des. Codes Cryptogr. 86, 1–16 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  8. Delsarte P.: Bilinear forms over a finire field, with applications to coding theory. J. Comb. Theory, Ser. A 25, 226–241 (1978).

    Article  MATH  Google Scholar 

  9. Gabidulin E.M.: Theory of codes with maximum rank distance (Russian). Probl. Pereda. Informatsii 21, 3–16 (1985).

    MathSciNet  MATH  Google Scholar 

  10. Gabidulin E.M.: Theory of codes with maximum rank distance (Translation). Probl. Inf. Transm. 1, 1–12 (1985).

    MathSciNet  MATH  Google Scholar 

  11. Gabidulin E.M., Paramonov A.V., Tretjakov O.V.: Ideals over a non-commutative ring and their application in cryptology. In: Proceedings of the Workshop on the Theory and Application of Cryptographic Techniques (EUROCRYPT 91), vol. 547 of Lecture Notes in Computer Science, pp. 482–489, Brighton, UK, (April 1991).

  12. Gadouleau M., Yan Z.: MacWilliams identity for the rank metric. In: ISIT2007, Nice, France, June 24–June 29 , pp. 36–40 (2007).

  13. Gadouleau M., Yan Z.: MacWilliams identity for the rank metric. EURASIP J. Wirel. Commun. Netw. https://doi.org/10.1155/2008/754021, (2008).

  14. Greene C.: Weight enumeration and the geometry of linear codes. Stud. Appl. Math. 55, 119–128 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  15. Huffman W., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).

    Book  MATH  Google Scholar 

  16. Koga Y., Maruta T., Shiromoto K.: On critical exponents of Dowling matroids. Des. Codes Cryptogr. 86, 1947–1962 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  17. Kshevetskiy A., Gabidulin E.N.: The new construction of rank codes. In: Proceedings of IEEE International Symposium on Information Theory (ISIT05), pp. 2105–2108, Adelaide, Australia, (2005).

  18. Lam T.Y.: Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189. Springer, New York (1999).

    Book  Google Scholar 

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

    MATH  Google Scholar 

  20. Oxley J.: Matroid Theory. Oxford University Press, New York (1992).

    MATH  Google Scholar 

  21. Oxley J., Whittle G.: A characterization of Tutte invariants of \(2\)-polymatroids. J. Comb. Theory Ser. B 59, 210–244 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  22. Ravagnani A.: Rank-metric codes and their duality theory. Des. Codes Cryptogr. 80, 197–216 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  23. Shiromoto K.: A basic exact sequence for the Lee and Euclidean weights of linear codes over \({\mathbb{Z}}_{\ell }\). Linear Algebra Its Appl. 295, 191–200 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  24. Silva D., Kschischang F.R., Koetter R.: A rank-metric approach to error control in random network coding. IEEE Trans. Inform. Theory 54, 3951–3967 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  25. Simonis J.: The effective length of subcodes. Appl. Algebra Eng. Commun. Comput. 5, 371–377 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  26. Simonis J., Ashikhmin A.: Almost affine codes. Des. Codes Cryptogr. 14, 179–197 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  27. Tarokh V., Seshadri N., Calderbank A.R.: Space-time codes for high data rate wireless communication: performance criterion and code construction. IEEE Trans. Inform. Theory 44, 744–765 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  28. Vertigan D.: Latroids and their representation by codes over modules. Trans. Am. Math. Soc. 356, 3841–3868 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  29. Welsh D.J.A.: Matroid Theory. Academic Press, London (1976).

    MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank Professor Frederique Oggier for introducing the concept of rank-metric codes. The author would like to thank the anonymous reviewers for their very helpful comments and suggestions that improved the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics and Engineering, Kumamoto University, 2-39-1, Kurokami, Kumamoto, 860-8555, Japan

    Keisuke Shiromoto

Authors
  1. Keisuke Shiromoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Keisuke Shiromoto.

Additional information

Communicated by C. Ding.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by JSPS KAKENHI Grant Number 17K05348.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shiromoto, K. Codes with the rank metric and matroids. Des. Codes Cryptogr. 87, 1765–1776 (2019). https://doi.org/10.1007/s10623-018-0576-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-018-0576-0

Keywords

Mathematics Subject Classification

Search

Navigation