Venkatesan Guruswami and Atri Rudra

Concatenated codes can achieve list-decoding capacity

Abstract :

We prove 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 the list-decoding capacity with high probability. 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 ρ of errors. (The best possible rate, aka list-decoding capacity, for such codes is 1-H(ρ), and is achieved by random codes.) A similar result, with better list size guarantees, holds when the outer code is also randomly chosen. Our methods and results extend to the case when the alphabet size is any fixed prime power q ≥ 2.

Our result shows that despite the structural restriction imposed by code concatenation, the family of concatenated codes is rich enough to include capacity achieving list-decodable codes. This provides some encouraging news for tackling the problem of constructing explicit binary list-decodable codes with optimal rate, since code concatenation has been the preeminent method for constructing good codes over small alphabets.