Efficiently decodable codes meeting Gilbert-Varshamov bound for low rates
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Algorithmic results in list decoding
Foundations and Trends® in Theoretical Computer Science
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We present a decoding algorithm for concatenated codes when the outer code is a Reed-Solomon code and the inner code is arbitrary. "Soft" information on the reliability of various symbols is passed by the inner decodings and exploited in the Reed-Solomon decoding. This is the first analysis of such a soft algorithm that works for arbitrary inner codes; prior analyses could only handle some special inner codes. Crucial to our analysis is a combinatorial result on the coset weight distribution of codes given only its minimum distance. Our result enables us to decode essentially up to the "Johnson radius" of a concatenated code when the outer distance is large (the Johnson radius is the "a priori list decoding radius" of a code as a function of its distance). As a consequence, we are able to present simple and efficient constructions of quary linear codes that are list decodable up to a fraction (1-{1 \over q}-\varepsilon ) of errors and have rate \Omega (\varepsilon ^6). Codes that can correct such a large fraction of errors have found numerous complexity-theoretic applications. The previous constructions of linear codes with a similar rate used algebraic-geometric codes and thus suffered from a complicated construction and slow decoding.