Improved List-Decodability and List-Recoverability of Reed-Solomon Codes via Tree Packings: [Extended Abstract]

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 E (0,1] there exist RS codes with rate Ω(1\not(1/in)+1}) that are list-decodable from radius of 1-ϵ. We generalize this result to list-recovery, showing that there exist (1-,ell, O(ℓ)-list-recoverable RS codes with rate Ω≤ft(√\ell}(log(1/)+1)right). 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. We also state a new conjecture that generalizes the tree-packing theorem to hypergraphs, and show that if this conjecture holds, then there would exist RS codes that are optimally (non-asymptotically) list-decodable.11A full version of this paper is available online at https://arxiv.org/abs/2011.04453.