Correcting errors beyond the Guruswami-Sudan radius in polynomial time

Abstract

We introduce a new family of error-correcting codes that have a polynomial-time encoder and a polynomial-time listdecoder, correcting a fraction of adversarial errors up to τ M = l - M+1√M MR M where R is the rate of the code and M ≥1 is an arbitrary integer parameter. This makes it possible to decode beyond the Guruswami-Sudan radius of 1 - √R for all rates less than 1/16. Stated another way, for any ε > 0, we can list-decode in polynomial time a fraction of errors up to 1 - ε with a code of length n and rate Ω(ε/log(1/ε)), defined over an alphabet of size n M = n O(log(1/ε)). Notably, this error-correction is achieved in the worst-case against adversarial errors: a probabilistic model for the error distribution is neither needed nor assumed. The best results so far for polynomial-time list-decoding of adversarial errors required a rate of of O(ε 2) to achieve the correction radius of 1 -ε. Our codes and list-decoders are based on two key ideas. The first is the transition from bivariate polynomial interpolation, pioneered by Sudan and Guruswami-Sudan [12,22], to multivariate interpolation decoding. The second idea is to pan ways with Reed-Solomon codes, for which numerous prior attempts [2,3,12,18] at breaking the O(ε 2) rate barrier in the worst-case were unsuccessful. Rather than devising a better list-decoder for Reed-Solomon codes, we devise better codes. Standard Reed-Solomon encoders view a message as a polynomial f (X) over a field double-stract F sign q, and produce the corresponding codeword by evaluating f (X) at n distinct elements of double-stract F sign q. Herein, given f (X), we first compute one or more related polynomials g 1 (X), g 2 (X),.. ./g M-1 (X) and produce the corresponding codeword by evaluating all these polynomials. Correlation between f (X) and g i (X), carefully designed into our encoder, then provides the additional information we need to recover the encoded message from the output of the multivariate interpolation process. © 2005 IEEE.