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.


Average Correlation Binary Code Block Length List Size Alphabet Size 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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

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