论文标题
改进的芦苇列表可调节性 - 通过树包装的固体代码
Improved List-Decodability of Reed--Solomon Codes via Tree Packings
论文作者
论文摘要
本文表明，存在芦苇 - 固体（rs）代码，超过\ black {指数}大的有限字段\ black {在代码长度}中，它们在约翰逊半径范围之外均可组合列表，实际上几乎可以实现列表编码的能力。 In particular, we show that for any $ε\in (0,1]$ there exist RS codes with rate $Ω(\fracε{\log(1/ε)+1})$ that are list-decodable from radius of $1-ε$. We generalize this result to list-recovery, showing that there exist $(1 - ε, \ell, O(\ell/ε))$-list-recoverable RS codes利率$ω\ left（\fracε{\ sqrt {\ ell}（\ log（1/ε）+1）} \ right）$。 为了得出本文的结果，我们显示了上述问题与图理论的令人惊讶的联系，尤其是纳什·威廉姆斯和tutte的树木包装定理。我们还指出了一个新的猜想，将树木包装定理推广到超图，并表明如果这种猜想成立，那么存在\ em em ther \ em em \ em（非征求力）列表可解码的RS代码。
This paper shows that there exist Reed--Solomon (RS) codes, over \black{exponentially} large finite fields \black{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 $ε\in (0,1]$ there exist RS codes with rate $Ω(\fracε{\log(1/ε)+1})$ that are list-decodable from radius of $1-ε$. We generalize this result to list-recovery, showing that there exist $(1 - ε, \ell, O(\ell/ε))$-list-recoverable RS codes with rate $Ω\left( \fracε{\sqrt{\ell} (\log(1/ε)+1)} \right)$. 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 \em optimally \em (non-asymptotically) list-decodable.
