[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

List decodability at small radii

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

Abstract

A′(n, d, e), the smallest for which every binary error-correcting code of length n and minimum distance d is decodable with a list of size up to radius e, is determined for all d ≥ 2e − 3. As a result, A′(n, d, e) is determined for all e ≤ 4, except for 42 values of n.

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

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Ahlswede R.: Channel capacities for list codes. J. Appl. Probab. 10(4), 824–836 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bassalygo L.A.: New upper bounds for error-correcting codes. Problemy Peredači Informacii 1(vyp. 4), 41–44 (1965)

    MathSciNet  MATH  Google Scholar 

  3. Blinovsky V.: Bounds for codes in the case of list decoding of finite volume. Probl. Inform. Transm. 22(1), 7–19 (1986)

    Google Scholar 

  4. Blinovsky V.: Asymptotic Combinatorial Coding Theory. The Kluwer International Series in Engineering and Computer Science, 415. Kluwer Academic Publishers, Boston (1997).

  5. Brouwer A.E.: On the packing of quadruples without common triples. Ars Combin. 5, 3–6 (1978)

    MathSciNet  MATH  Google Scholar 

  6. Brouwer A.E.: Optimal packings of K 4’s into a K n . J. Comb. Theory A 26(3), 278–297 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brouwer A.E., Shearer J.B., Sloane N.J.A., Smith W.D.: A new table of constant weight codes. IEEE Trans. Inform. Theory 36(6), 1334–1380 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cassuto Y., Bruck J.: A combinatorial bound on the list size. Technical Report, California Institute of Technology, Pasedna, CA (2004).

  9. Elias P.: List decoding for noisy channels. Technical report 335. Research Laboratory of Electronics, Massachusetts Institute of Technology (1957).

  10. Elias P.: Error-correcting codes for list decoding. IEEE Trans. Inform. Theory 37(1), 5–12 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Forney G.D.: Exponential error bounds for erasure, list and decision feedback schemes. IEEE Trans. Inform. Theory 14, 549–557 (1968)

    Article  MathSciNet  Google Scholar 

  12. Goldreich O., Levin L.A.: A hard-core predicate for all one-way functions. In: STOC 1989: Proceedings of the 21st Annual ACM Symposium on Theory of Computing, 15–17 May 1989, Seattle, Washington, USA, pp. 25–32. ACM Press (1989).

  13. Guruswami V.: List Decoding of Error-Correcting Codes, Lecture Notes in Computer Science, vol. 3282. Springer-Verlag, Berlin (2004).

  14. Guruswami V., Sudan M.: Extensions to the Johnson bound. Unpublished manuscript (2001).

  15. Hanani H.: On quadruple systems. Can. J. Math. 12, 145–157 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ji L.: Asymptotic determination of the last packing number of quadruples. Des. Codes Cryptogr. 38(1), 83–95 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Johnson S.M.: Upper bounds for constant weight error-correcting codes. Discrete Math. 3, 109–124 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schönheim J.: On maximal systems of k-tuples. Studia Sci. Math. Hungar 1, 363–368 (1966)

    MathSciNet  MATH  Google Scholar 

  19. Shannon C.E., Gallager R.G., Berlekamp E.R.: Lower bounds to error probability for coding on discrete memoryless channels. I. Inform. Control 10, 65–103 (1967)

    Article  MathSciNet  Google Scholar 

  20. Shannon C.E., Gallager R.G., Berlekamp E.R.: Lower bounds to error probability for coding on discrete memoryless channels. II. Inform. Control 10, 522–552 (1967)

    Article  MathSciNet  Google Scholar 

  21. Spencer J.: Maximal consistent families of triples. J. Comb. Theory 5, 1–8 (1968)

    Article  MATH  Google Scholar 

  22. Sudan M.: Decoding of Reed Solomon codes beyond the error-correction bound. J. Complex. 13(1), 180–193 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wozencraft J.M.: List decoding. Research Laboratory of Electronics, Massachusetts Institute of Technology, Progress report 48, pp. 90–95 (1958).

  24. Zyablov V.V., Pinsker M.S.: List cascade decoding. Probl. Inform. Transm. 17(4), 236–240 (1982)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yeow Meng Chee.

Additional information

Communicated by V. A. Zinoviev.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chee, Y.M., Ge, G., Ji, L. et al. List decodability at small radii. Des. Codes Cryptogr. 61, 151–166 (2011). https://doi.org/10.1007/s10623-010-9445-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-010-9445-1

Keywords

Mathematics Subject Classification (2000)