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

Abstract

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/ε) bits that can be list decoded from a fraction (1/2 − ε) of errors (even for ε= o(1)) in poly(k/ε) 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/ε, whereas the existential bound is n = O(k/ε ²). (The above-mentioned result decoding up to Zyablov radius achieves a rate of Ω(ε ³) for the case of constant ε.)

We will only sketch the high level ideas behind these developments, pointing to the original papers for technical details and precise theorem statements.