Unique Decoding of Reed-Solomon and Related Codes for Semi-Adversarial Errors

Abstract: For over a quarter century, the Guruswami--Sudan algorithm has served as the state-of-the-art for list-decoding Reed--Solomon (RS) codes up to the Johnson bound against adversarial errors. However, some recent structural results on the combinatorial list decoding of randomly punctured Reed--Solomon codes suggest that Johnson bound can likely be broken for some subclasses of RS codes. Motivated by these results, we seek to make traction on understanding adversarial decoding by considering a new model: semi-adversarial errors. This error model bridges between fully random errors and fully adversarial errors by allowing some symbols of a message to be corrupted by an adversary while others are replaced with uniformly random symbols. As our main quest, we seek to understand optimal efficient unique decoding algorithms in the semi-adversarial model. In particular, we revisit some classical results on decoding interleaved Reed--Solomon codes (aka subfield evaluation RS codes) in the random error model by Bleichenbacher--Kiayias--Yung (BKY) and work to improve and extend their analysis. First, we give an improved implementation and analysis of the BKY algorithm for interleaved Reed--Solomon codes in the semi-adversarial model. In particular, our algorithm runs in near-linear time, and for most mixtures of random and adversarial errors, our analysis matches the information-theoretic optimum. Moreover, inspired by the BKY algorithm, we use a novel interpolation to extend our approach to the settings of folded Reed--Solomon and multiplicity codes, resulting in fast algorithms for unique decoding against semi-adversarial errors. A particular advantage of our near-linear time algorithm over state-of-the-art decoding algorithms for adversarial errors is that its running time depends only on a polynomial function of the folding parameter rather than on an exponential function.