List-decodability with large radius for Reed-Solomon codes
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List-decodability of Reed-Solomon codes has received a lot of attention by different researchers, but the best possible dependence between the parameters is still not well-understood. In this work, we focus on the case where the list-decoding radius is of the form r=1-ε for ε tending to zero. Our main result states that there exist (1-ε, O(1/ε))-list-decodable Reed-Solomon codes with rate Ω(ε) (which is best-possible for any code which is list-decodable with radius 1-ε and list size less than exponential in the block length). This improves a recent result of Guo, Li, Shangguan, Tamo, and Wootters, and resolves the main motivating question of their work. We deduce our main result from a more general theorem, in which we prove good list-decodability properties of random puncturings of a given code with very large distance.
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