A Lower Bound on List Size for List Decoding

  • Venkatesan Guruswami
  • Salil Vadhan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3624)

Abstract

A q-ary error-correcting code C ⊆ {1,2,...,q} n is said to be list decodable to radius ρ with list size L if every Hamming ball of radius ρ contains at most L codewords of C. We prove that in order for a q-ary code to be list-decodable up to radius (1–1/q)(1–ε)n, we must have L = Ω(1/ε 2). Specifically, we prove that there exists a constant c q >0 and a function f q such that for small enough ε > 0, if C is list-decodable to radius (1–1/q)(1–ε)n with list size c q /ε 2, then C has at most f q (ε) codewords, independent of n. This result is asymptotically tight (treating q as a constant), since such codes with an exponential (in n) number of codewords are known for list size L = O(1/ε 2).

A result similar to ours is implicit in Blinovsky [Bli] for the binary (q=2) case. Our proof works for all alphabet sizes, and is technically and conceptually simpler.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bli]
    Blinovsky, V.M.: Bounds for codes in the case of list decoding of finite volume. Problems of Information Transmission 22(1), 7–19 (1986)Google Scholar
  2. [Eli1]
    Elias, P.: List decoding for noisy channels. Technical Report 335, Research Laboratory of Electronics, MIT (1957)Google Scholar
  3. [Eli2]
    Elias, P.: Error-correcting codes for list decoding. IEEE Transactions on Information Theory 37, 5–12 (1991)MATHCrossRefMathSciNetGoogle Scholar
  4. [Gur]
    Guruswami, V.: List decoding from erasures: Bounds and code constructions. IEEE Transactions on Information Theory 49(11), 2826–2833 (2003)CrossRefMathSciNetGoogle Scholar
  5. [GHSZ]
    Guruswami, V., Hastad, J., Sudan, M., Zuckerman, D.: Combinatorial bounds for list decoding. IEEE Transactions on Information Theory 48(5), 1021–1035 (2002)MATHCrossRefMathSciNetGoogle Scholar
  6. [LTW]
    Lu, C.-J., Tsai, S.-C., Wu, H.-L.: On the complexity of hardness amplification. In: Proceedings of the 20th Annual IEEE Conference on Computational Complexity, San Jose, CA (June 2005) (to appear)Google Scholar
  7. [RT]
    Radhakrishnan, J., Ta-Shma, A.: Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM Journal on Discrete Mathematics 13(1), 2–24 (2000) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  8. [TZ]
    Ta-Shma, A., Zuckerman, D.: Extractor codes. IEEE Transactions on Information Theory 50(12), 3015–3025 (2004)CrossRefMathSciNetGoogle Scholar
  9. [Tre]
    Trevisan, L.: Extractors and Pseudorandom Generators. Journal of the ACM 48(4), 860–879 (2001)MATHCrossRefMathSciNetGoogle Scholar
  10. [Vad]
    Vadhan, S.P.: Randomness Extractors and their Many Guises. Tutorial at IEEE Symposium on Foundations of Computer Science (November 2002), Slides available at http://eecs.harvard.edu/~salil
  11. [Woz]
    Wozencraft, J.M.: List Decoding. Quarterly Progress Report, Research Laboratory of Electronics 48, 90–95 (1958)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Venkatesan Guruswami
    • 1
  • Salil Vadhan
    • 2
  1. 1.Department of Computer Science & EngineeringUniversity of WashingtonSeattle
  2. 2.Division of Engineering & Applied SciencesHarvard UniversityCambridge

Personalised recommendations