Two theorems on list decoding

• Authors: Atri Rudra;Steve Uurtamo
• Venue: APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
• Year: 2010

We prove the following results concerning the list decoding of error-correcting codes: 1. We show that for any code with a relative distance of δ (over a large enough alphabet), the following result holds for random errors: With high probability, for a ρ ≤ δ - ε fraction of random errors (for any ε 0), the received word will have only the transmitted codeword in a Hamming ball of radius ρ around it. Thus, for random errors, one can correct twice the number of errors uniquely correctable from worst-case errors for any code. A variant of our result also gives a simple algorithm to decode Reed-Solomon codes from random errors that, to the best of our knowledge, runs faster than known algorithms for certain ranges of parameters. 2. We show that concatenated codes can achieve the list decoding capacity for erasures. A similar result for worst-case errors was proven by Guruswami and Rudra (SODA 08), although their result does not directly imply our result. Our results show that a subset of the random ensemble of codes considered by Guruswami and Rudra also achieve the list decoding capacity for erasures. We also show that the exponential list size bound in our result with outer random linear codes cannot be improved using the recent techniques of Guruswami, Håstad and Kopparty that achieved similar improvements for errors. Our proofs employ simple counting and probabilistic arguments.