List Decoding of Binary Codes---A Brief Survey of Some Recent Results

• Authors: Venkatesan Guruswami
• Venue: IWCC '09 Proceedings of the 2nd International Workshop on Coding and Cryptology
• Year: 2009

We briefly survey some recent progress on list decoding algorithms for binary codes. The results discussed include: Algorithms to list decode binary Reed-Muller codes of any order up to the minimum distance, generalizing the classical Goldreich-Levin algorithm for RM codes of order 1 (Hadamard codes). These algorithms are "local" and run in time polynomial in the message length. Construction of binary codes efficiently list-decodable up to the Zyablov (and Blokh-Zyablov) radius. This gives a factor two improvement over the error-correction radius of traditional "unique decoding" algorithms. The existence of binary linear concatenated codes that achieve list decoding capacity, i.e., the optimal trade-off between rate and fraction of worst-case errors one can hope to correct. Explicit binary codes mapping k bits to n ≤ poly(k /*** ) bits that can be list decoded from a fraction (1/2 *** *** ) of errors (even for *** = o (1)) in poly(k /*** ) time. A construction based on concatenating a variant of the Reed-Solomon code with dual BCH codes achieves the best known (cubic) dependence on 1/*** , whereas the existential bound is n = O (k /*** 2). (The above-mentioned result decoding up to Zyablov radius achieves a rate of *** (*** 3) for the case of constant *** .) We will only sketch the high level ideas behind these developments, pointing to the original papers for technical details and precise theorem statements.