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On the List-Decodability of Random Linear Codes

By Venkatesan Guruswami

Abstract

Abstract—The list-decodability of random linear codes is shown to be as good as that of general random codes. Specifically, for every fixed finite field Fq, p ∈ (0, 1 − 1/q) and ε>0, itis proved that with high probability a random linear code C in F n q of rate (1 − Hq(p) − ε) can be list decoded from a fraction p of errors with lists of size at most O(1/ε). This also answers a basic open question concerning the existence of highly list-decodable linear codes, showing that a listsize of O(1/ε) suffices to have rate within ε of the informationtheoretically optimal rate of 1−Hq(p). The best previously known list-size bound was q O(1/ε) (except in the q =2case where a listsize bound of O(1/ε) was known). The main technical ingredient in the proof is a strong upper bound on the probability that ℓ random vectors chosen from a Hamming ball centered at the origin have too many (more than Ω(ℓ)) vectors from their linear span also belong to the ball

Topics: List decoding, Random coding, Probabilistic method, Hamming bound
Year: 2011
OAI identifier: oai:CiteSeerX.psu:10.1.1.185.1341
Provided by: CiteSeerX
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