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

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Omega(\frac{\varepsilon}{1\not\varepsilon(1/_{\in})+1})$</tex> that are list-decodable from radius of 1-ε. We generalize this result to list-recovery, showing that there exist <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(1-\varepsilon,\ell, O(\ell/\varepsilon))$</tex> -list-recoverable RS codes with rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Omega\left(\frac{\varepsilon}{\sqrt{\ell}(\log(1/\varepsilon)+1)}\right)$</tex> . 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. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> A full version of this paper is available online at https://arxiv.org/abs/2011.04453.