5 research outputs found
List Decoding of Matrix-Product Codes from nested codes: an application to Quasi-Cyclic codes
A list decoding algorithm for matrix-product codes is provided when are nested linear codes and is a non-singular by columns matrix. We
estimate the probability of getting more than one codeword as output when the
constituent codes are Reed-Solomon codes. We extend this list decoding
algorithm for matrix-product codes with polynomial units, which are
quasi-cyclic codes. Furthermore, it allows us to consider unique decoding for
matrix-product codes with polynomial units
Two Theorems in List Decoding
We prove the following results concerning the list decoding of
error-correcting codes:
(i) We show that for \textit{any} code with a relative distance of
(over a large enough alphabet), the following result holds for \textit{random
errors}: With high probability, for a \rho\le \delta -\eps fraction of random
errors (for any \eps>0), the received word will have only the transmitted
codeword in a Hamming ball of radius around it. Thus, for random errors,
one can correct twice the number of errors uniquely correctable from worst-case
errors for any code. A variant of our result also gives a simple algorithm to
decode Reed-Solomon codes from random errors that, to the best of our
knowledge, runs faster than known algorithms for certain ranges of parameters.
(ii) We show that concatenated codes can achieve the list decoding capacity
for erasures. A similar result for worst-case errors was proven by Guruswami
and Rudra (SODA 08), although their result does not directly imply our result.
Our results show that a subset of the random ensemble of codes considered by
Guruswami and Rudra also achieve the list decoding capacity for erasures.
Our proofs employ simple counting and probabilistic arguments.Comment: 19 pages, 0 figure
Better binary list-decodable codes via multilevel concatenation
Abstract. 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 âlevel-by-level â 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