Advertisement

Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 481–493 | Cite as

A distance between channels: the average error of mismatched channels

  • Rafael G. L. D’Oliveira
  • Marcelo FirerEmail author
Article
  • 67 Downloads
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography

Abstract

Two channels are equivalent if their maximum likelihood (ML) decoders coincide for every code. We show that this equivalence relation partitions the space of channels into a generalized hyperplane arrangement. With this, we define a coding distance between channels in terms of their ML-decoders which is meaningful from the decoding point of view, in the sense that the closer two channels are, the larger is the probability of them sharing the same ML-decoder. We give explicit formulas for these probabilities.

Keywords

Mismatched channels Maximum likelihood decoding Space of channels 

Mathematics Subject Classification

68P30 51E22 52C35 

Notes

Acknowledgements

Rafael G. L. D’Oliveira was supported by CAPES. Marcelo Firer was partially supported by São Paulo Research Foundation, (FAPESP Grant 2013/25977-7).

References

  1. 1.
    Deza M., Deza E.: Encyclopedia of Distances, 4th revised edition. Springer, Dordrecht (2016).Google Scholar
  2. 2.
    D’Oliveira R.G.L., Firer M.: Geometry of communication channels: metrization and decoding. Symmetry Cult. Sci. 27(4), 279–289 (2016).Google Scholar
  3. 3.
    D’Oliveira R.G.L., Firer M.: Minimum dimensional Hamming embeddings. Adv. Math. Commun. 11(2), 359–366 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D’Oliveira R.G.L., Firer M.: Channel metrization. Eur. J. Comb. (2018).  https://doi.org/10.1016/j.ejc.2018.02.026.
  5. 5.
    Firer M., Walker J.L.: Matched metrics and channels. IEEE Trans. Inf. Theory 62(3), 1150–1156 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gabidulin E.: A brief survey of metrics in coding theory. In: Mathematics of Distances and Applications, pp. 66–84 (2012).Google Scholar
  7. 7.
    Ganti A., Lapidoth A., Telatar E.: Mismatched decoding revisited: general alphabets, channels with memory, and the wide-band limit. IEEE Trans. Inf. Theory 46(7), 2315–2328 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kendall M.: A new measure of rank correlation. Biometrika 30, 81–89 (1938).CrossRefzbMATHGoogle Scholar
  9. 9.
    Makur A., Polyanskiy Y.: Comparison of channels: criteria for domination by a symmetric channel. IEEE Trans. Inf. Theory 64(8), 5704–5725 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Poplawski A.: On Matched Metric and Channel Problem. arXiv:1606.02763 [cs.IT] (2016).
  11. 11.
    Qureshi C.: Matched Metrics to the Binary Asymmetric Channels. arXiv:1606.09494 [cs.IT] (2016).
  12. 12.
    Scarlett J., Martinez A.: Fabregas A.G.i.: Mismatched decoding: error exponents, second-order rates and saddlepoint approximations. IEEE Trans. Inf. Theory 60(5), 2647–2666 (2014).CrossRefzbMATHGoogle Scholar
  13. 13.
    Séguin G.: On metrics matched to the discrete memoryless channel. J. Frankl. Inst. 309(3), 179–189 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shannon C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shannon C.E.: A note on a partial ordering for communication channels. Inf. Control 1, 390–397 (1958).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Stanley R.P.: An Introduction to Hyperplane Arrangements. Lecture Notes. IAS/Park City Mathematics Institute, Park City (2004).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Rutgers UniversityNew BrunswickUSA
  2. 2.University of CampinasCampinasBrazil

Personalised recommendations