IMPROVED LIST-DECODABILITY AND LIST-RECOVERABILITY OF REED-SOLOMON CODES VIA TREE PACKINGS

This paper shows that there exist Reed-Solomon (RS) codes, over exponentially large finite fields in the code length, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving the list-decoding capacity. In particular, we show that for any \varepsilon \in (0, 1] there exist RS codes with rate (Formula presented) that are list-decodable from radius of 1 - \varepsilon. We generalize this result to list-recovery, showing that there exist (1 - \varepsilon, \ell, O(\ell/\varepsilon))-list-recoverable RS codes with rate (Formula presented). 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 (nonasymptotically) list-decodable.