[Paper Review] Improved List-Decodability of Reed--Solomon Codes via Tree Packings

Zeyu Guo, Ray Li+3 more|arXiv (Cornell University)|Nov 9, 2020

Coding theory and cryptography35 references4 citations

TL;DR

This paper demonstrates that Reed–Solomon codes over exponentially large fields can achieve near-capacity list-decoding and list-recovery beyond the Johnson radius by leveraging a novel connection to tree packing theorems in graph theory. It proves the existence of RS codes with rate Ω(ε/log(1/ε)) list-decodable up to radius 1−ε, and Ω(ε/(√ℓ(log(1/ε)+1))) rate for (1−ε,ℓ,O(ℓ/ε))-list-recoverable codes, establishing a strong link between coding theory and combinatorial graph structures.

ABSTRACT

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 $Ω\left( \fracε{\sqrt{\ell} (\log(1/ε)+1)} ight)$. 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.

Motivation & Objective

• To improve the known list-decoding and list-recovery capabilities of Reed–Solomon codes beyond the Johnson radius.
• To establish a novel connection between coding theory and graph theory, specifically the Nash-Williams–Tutte tree-packing theorem.
• To prove the existence of RS codes with near-optimal rate for high-radius list-decoding and list-recovery, approaching the theoretical capacity limit.
• To generalize results on perfect hash matrices and strengthen them to construct strongly perfect hash matrices.
• To propose a hypergraph generalization of the Nash-Williams–Tutte theorem, which, if true, would yield optimally list-decodable RS codes.

Proposed method

• Utilizes the Nash-Williams–Tutte tree-packing theorem to construct codes with high list-decoding radius by analyzing cycle spaces and t-wise intersection matrices.
• Employs combinatorial matrix constructions, including variable matrices and block matrices, to model codeword intersections and ensure list-recovery properties.
• Applies the theory of $t$-wise intersection matrices to bound the number of codewords within a Hamming ball, enabling list-decoding guarantees.
• Reduces the problem of list-decoding and list-recovery to hypergraph packing problems, introducing a new conjecture generalizing the tree-packing theorem to hypergraphs.
• Uses a duality between perfect hashing matrices and list-recoverable codes to derive new bounds on code rates and list sizes.
• Proves a new result on optimal linear perfect hash matrices by reinterpreting them via the tree-packing framework, strengthening it to strongly perfect hash matrices.

Experimental results

Research questions

• RQ1 Can Reed–Solomon codes be list-decoded beyond the Johnson radius, approaching the list-decoding capacity?
• RQ2 What is the maximal rate of RS codes that are list-decodable up to radius 1−ε for any ε∈(0,1]?
• RQ3 How can the tree-packing theorem of Nash-Williams and Tutte be leveraged to improve code constructions in coding theory?
• RQ4 Can the connection between perfect hash matrices and list-recoverable codes be used to derive stronger existential bounds on code rates?
• RQ5 Does a hypergraph generalization of the Nash-Williams–Tutte theorem exist, and what would its implications be for optimal list-decoding of RS codes?

Key findings

• There exist Reed–Solomon codes with rate Ω(ε/log(1/ε)) that are list-decodable from radius 1−ε, significantly exceeding the Johnson bound.
• For list-recovery, the paper constructs (1−ε,ℓ,O(ℓ/ε))-list-recoverable RS codes with rate Ω(ε/(√ℓ(log(1/ε)+1))), improving over prior generic bounds.
• The authors establish a new proof of Blackburn’s result on optimal linear perfect hash matrices and strengthen it to construct strongly perfect hash matrices.
• A surprising connection is revealed between list-decoding of RS codes and the Nash-Williams–Tutte tree-packing theorem, enabling new existential constructions.
• The paper formulates a new conjecture generalizing the tree-packing theorem to hypergraphs, which, if true, would yield optimally list-decodable RS codes.
• The results demonstrate that the list-decoding capacity of RS codes is not a barrier for existence, even if efficient algorithms remain open.