Improved List-Decodability of Reed–Solomon Codes via Tree Packings
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This paper shows that there exist Reed–Solomon (RS) codes, over large finite fields, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving list-decoding capacity. In particular, we show that for any ϵ∈ (0,1] there exist RS codes with rate Ω(ϵ/log(1/ϵ)+1) that are list-decodable from radius of 1-ϵ. We generalize this result to obtain a similar result on list-recoverability of RS codes. Along the way we use our techniques to give a new proof of a result of Blackburn on optimal linear perfect hash matrices, and strengthen it to obtain a construction of strongly perfect hash matrices. To derive the results in this paper we show a surprising connection of the above problems to graph theory, and in particular to the tree packing theorem of Nash-Williams and Tutte. En route to our results on RS codes, we prove a generalization of the tree packing theorem to hypergraphs (and we conjecture that an even stronger generalization holds). We hope that this generalization to hypergraphs will be of independent interest.
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