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)


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.


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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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