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### Paper:

TR10-007 | 12th January 2010 15:16

#### Two Theorems in List Decoding

TR10-007
Authors: Atri Rudra, steve uurtamo
Publication: 14th January 2010 13:59
Keywords:

Abstract:

We prove the following results concerning the list decoding of error-correcting codes:

We show that for any code with a relative distance of $\delta$
(over a large enough alphabet), the
following result holds for random errors: With high probability,
for a $\rho\le \delta -\eps$ fraction of random errors (for any $\eps>0$),
the received word will have only the transmitted codeword in a Hamming ball of
radius $\rho$ 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.

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.

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