We give a polynomial time construction of binary codes with the best currently known trade-off between rate and error-correction radius. Specifically, we obtain linear codes over fixed alphabets that can be list decoded in polynomial time up to the so called Blokh-Zyablov bound. Our work builds upon [7] where codes list decodable up to the Zyablov bound (the standard product bound on distance of concatenated codes) were constructed. Our codes are constructed via a (known) generalization of code concatenation called multilevel code concatenation. A probabilistic argument, which is also derandomized via conditional expectations, is used to show the existence of inner codes with a certain nested list decodability property that is appropriate for use in multilevel concatenated codes. A "level-by-level" decoding algorithm, which crucially uses the list recovery algorithm for folded Reed-Solomon codes from [7], enables list decoding up to the designed distance bound, aka the Blokh-Zyablov bound, for multilevel concatenated codes.
[1]
Venkatesan Guruswami,et al.
Expander-based constructions of efficiently decodable codes
,
2001,
Proceedings 2001 IEEE International Conference on Cluster Computing.
[2]
Noga Alon,et al.
The Probabilistic Method
,
2015,
Fundamentals of Ramsey Theory.
[3]
V. A. Zynovev.
Generalized concatenated code
,
1976
.
[4]
Venkatesan Guruswami,et al.
Explicit capacity-achieving list-decodable codes
,
2005,
STOC.
[5]
A. Rudra,et al.
List decoding and property testing of error-correcting codes
,
2007
.
[6]
Venkatesan Guruswami,et al.
List Decoding of Error-Correcting Codes (Winning Thesis of the 2002 ACM Doctoral Dissertation Competition)
,
2005,
Lecture Notes in Computer Science.