Improved List-Decodability and List-Recoverability of Reed–Solomon Codes via Tree Packings

IF 1.2 3区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

SIAM Journal on Computing Pub Date : 2024-03-19 DOI:10.1137/21m1463707

Zeyu Guo, Ray Li, Chong Shangguan, Itzhak Tamo, Mary Wootters

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引用次数: 0

Abstract

SIAM Journal on Computing, Volume 53, Issue 2, Page 389-430, April 2024.
Abstract. 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 [math] there exist RS codes with rate [math] that are list-decodable from radius of [math]. We generalize this result to list-recovery, showing that there exist [math]-list-recoverable RS codes with rate [math]. 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.

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通过树形包装提高里德-所罗门码的列表可解码性和列表可恢复性

SIAM 计算期刊》，第 53 卷第 2 期，第 389-430 页，2024 年 4 月。摘要本文表明，在码长为指数大的有限域上存在里德-所罗门（Reed-Solomon，RS）码，其组合列表可解码性远远超过约翰逊半径，事实上几乎达到了列表解码能力。特别是，我们证明了对于任何 [math]，都存在速率为 [math] 的 RS 码，这些码从 [math] 半径开始是可列表解码的。我们将这一结果推广到列表恢复，证明存在速率为[math]的[math]-列表可恢复 RS 编码。在此过程中，我们利用我们的技术对布莱克本关于最优线性完美哈希矩阵的一个结果给出了新的证明，并加强了这个结果，从而获得了强完美哈希矩阵的构造。为了得出本文的结果，我们展示了上述问题与图论，特别是与纳什-威廉姆斯和图特的树包装定理之间的惊人联系。我们还提出了一个新的猜想，将树打包定理推广到超图，并证明如果这个猜想成立，那么就会存在最佳（非渐近）列表可解码的 RS 编码。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Computing 工程技术-计算机：理论方法

CiteScore

4.60

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Computing aims to provide coverage of the most significant work going on in the mathematical and formal aspects of computer science and nonnumerical computing. Submissions must be clearly written and make a significant technical contribution. Topics include but are not limited to analysis and design of algorithms, algorithmic game theory, data structures, computational complexity, computational algebra, computational aspects of combinatorics and graph theory, computational biology, computational geometry, computational robotics, the mathematical aspects of programming languages, artificial intelligence, computational learning, databases, information retrieval, cryptography, networks, distributed computing, parallel algorithms, and computer architecture.