The existence of concatenated codes list-decodable up to the hamming bound

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, with high probability, the optimal trade-off between rate and list-decoding radius. In particular, for any 0 < < 1/2 and > 0, there exist concatenated codes of rate at least 1 - H() 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(), 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. 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 ...

• Type: Journal
• Year: 2010
• Where: TIT
• Authors: Venkatesan Guruswami, Atri Rudra
• Keywords: Code, Concatenated Codes, Education, Linear Concatenated Codes, TIT 2010