The Existence of Concatenated Codes List-Decodable up to the Hamming Bound

It is proven that binary linear concatenated codes with an outer algebraic code (specifically, a folded Reed-Solomon code) and independently and randomly chosen linear inner codes achieve, with high probability, the optimal tradeoff between rate and list-decoding radius. In particular, for any 0 <; ρ <; 1/2 and ε > 0, there exist concatenated codes of rate at least 1-H(ρ)-ε that are (combinatorially) list-decodable up to a fraction of errors. (The Hamming bound states that the best possible rate for such codes cannot exceed 1-H(ρ), and standard random coding arguments show that this bound is approached by random codes with high probability.) A similar result, with better list size guarantees, holds when the outer code is also randomly chosen. The methods and results extend to the case when the alphabet size is any fixed prime power q ≥ 2.