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Combinatorial metrics: MacWilliams-type identities, isometries and extension property

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Abstract

In this work we characterize the combinatorial metrics admitting a MacWilliams-type identity and describe the group of linear isometries of such metrics. Considering the binary case, we classify the metrics satisfying the MacWilliams extension property (for disconnected coverings) and give a necessary condition for the extension property (for connected coverings).

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References

  1. Bossert M., Sidorenko V.: Singleton-type bounds for blot-correcting codes. IEEE Trans. Inf. Theory 42(3), 1021–1023 (1996).

    Article  MATH  Google Scholar 

  2. Bridwell J.D., Wolf J.K.: Burst distance and multiple-burst correction. Bell Syst. Tech. J. 49(5), 889–909 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  3. Etzion, T., Firer, M., Machado, R.A.: Metrics based on finite directed graphs and coding invariants. arXiv preprint arXiv:1609.08067, (2016).

  4. Feng K., Lanju X., Hickernell F.J.: Linear error-block codes. Finite Fields Their Appl. 12(4), 638–652 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  5. Gabidulin, E.M.: Combinatorial metrics in coding theory. In: 2nd International Symposium on Information Theory. Akadémiai Kiadó, (1973).

  6. Gabidulin, E.: A brief survey of metrics in coding theory. Math. Distances Appl. 66 1–19 (2012).

  7. MacWilliams F.J.: Combinatorial Properties of Elementary Abelian Groups. Radcliffe College, Cambridge (1962).

    Google Scholar 

  8. MacWilliams J.: A theorem on the distribution of weights in a systematic code. Bell Syst. Tech. J. 42(1), 79–94 (1963).

    Article  MathSciNet  Google Scholar 

  9. Mohamed, M.H., Bossert, M.: Combinatorial metrics and collaborative error/erasure decoding for translational metrics. In: Information Technology and Systems (ITaS), pp. 550–559 (2015).

  10. Panek L., Firer M., Kim H.K., Hyun J.Y.: Groups of linear isometries on poset structures. Discret. Math. 308(18), 4116–4123 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  11. Pinheiro J.A., Firer M.: Classification of poset-block spaces admitting macwilliams-type identity. IEEE Trans. Inf. Theory 58(12), 7246–7252 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  12. Shi M., Shiromoto K., Solé P.: A note on a basic exact sequence for the lee and euclidean weights of linear codes over \(z_l\). Linear Algebra Its Appl. 475, 151–153 (2015).

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the São Paulo Research Foundation (Fapesp) for the financial support through three Grants: 2013/25977-7, 2017/14616-4 and 2017/10018-5.

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Authors and Affiliations

  1. Institute of Mathematics, Statistics and Scientific Computing, University of Campinas, Campinas, SP, Brazil

    Jerry Anderson Pinheiro, Roberto Assis Machado & Marcelo Firer

Authors
  1. Jerry Anderson Pinheiro
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  2. Roberto Assis Machado
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  3. Marcelo Firer
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Correspondence to Marcelo Firer.

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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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Pinheiro, J.A., Machado, R.A. & Firer, M. Combinatorial metrics: MacWilliams-type identities, isometries and extension property . Des. Codes Cryptogr. 87, 327–340 (2019). https://doi.org/10.1007/s10623-018-0527-9

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  • DOI: https://doi.org/10.1007/s10623-018-0527-9

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