A lower bound on list size for list decoding

  • Authors:

  • Venkatesan Guruswami;Salil Vadhan

  • Affiliations:

  • Department of Computer Science & Engineering, University of Washington, Seattle, WA;Division of Engineering & Applied Sciences, Harvard University, Cambridge, MA

  • Venue:
  • APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
  • Year:
  • 2005

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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/ε2). Specifically, we prove that there exists a constant cq0 and a function fq such that for small enough ε 0, if C is list-decodable to radius (1–1/q)(1–ε)n with list size cq/ε2, then C has at most fq(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/ε2). 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.