A New Upper Bound on Codes Decodable into Size-2 Lists

  • Alexei Ashikhmin
  • Alexander Barg
  • Simon Litsyn


A new asymptotic upper bound on the size of binary codes with the property described in the title is derived. The proof relies on the properties of the distance distribution of binary codes established in earlier related works of the authors.


Binary Code Distance Distribution List Code Code DECODABLE Error Exponent 
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  1. [1]
    R. Ahlswede, “Channel capacities for list codes”, J. Appl. Probability, 10, 1973, 824–836.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    A. Ashikhmin and A. Barg, `Binomial moments of the distance distribution: Bounds and applications“, IEEE Trans. Inform. Theory, 45, 1999, 438–452.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    A. Ashikhmin, A. Barg, and S. Litsyn, “New upper bounds on generalized distances”, IEEE Trans. Inform. Theory, 45, 1999, 1258–1263.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    V. Blinovsky, “Bounds for codes decodable in a list of finite size”, Problems of Information Transmission, 22 (1), 1986, 11–25.Google Scholar
  5. [5]
    V. Blinovsky, “Asymptotic Combinatorial Coding Theory”, Kluwer Academic Publishers, Boston, 1997.zbMATHCrossRefGoogle Scholar
  6. [6]
    P. Elias, “List decoding for noisy channels”, Rep. No. 335 Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Mass. MR 20 #5702, 1957.Google Scholar
  7. [7]
    P. Elias, “Error correcting codes for list decoding”, IEEE Trans. Inform. Theory, 37, 1991, 5–12.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    V. I. Levenshtein, “Upper-bound estimates for fixed-weight codes”, Problemy Peredachi Informatsii,7(4), 1971, 3–12, in Russian. English translation in Probl. Inform. Trans. 7, 281–287.Google Scholar
  9. [9]
    R. J. McEliece, E. R. Rodemich, H. Rumsey, and L. R. Welch, “New upper bound on the rate of a code via the Delsarte-MacWilliams inequalities”, IEEE Trans. Inform. Theory, 23, 1977, 157–166.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    S. Litsyn, “New bounds on error exponents”, IEEE Trans. Inform. Theory, 45, 1999, 385–398.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    A. Samorodnitsky, “On the optimum of Delsarte’s linear program”, J. Combinatorial Theory, Ser. A, to appear, 1999.Google Scholar
  12. [12]
    J. M. Wozencraft, “List decoding”, Quarterly Progr. Rep., Res. Lab. Electronics, MIT, 48, 1958, 90–95.Google Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Alexei Ashikhmin
    • 1
  • Alexander Barg
    • 2
  • Simon Litsyn
    • 3
  1. 1.Los Alamos National LaboratoryLos AlamosUSA
  2. 2.Bell LaboratoriesLucent TechnologiesMurray HillUSA
  3. 3.Department of Electrical Engineering-SystemsTel Aviv UniversityRamat AvivIsrael

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