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Generic Reed-Solomon Codes Achieve List-decoding Capacity

Joshua Brakensiek, Sivakanth Gopi, Visu Makam

TL;DR

The paper shows that several natural extensions of MDS codes—MDS(ℓ), LD-MDS(≤L), and GZP(ℓ)—are equivalent, up to duality, and that generic Reed-Solomon codes achieve list-decoding capacity for all rates. Central to the results is a dimension formula for generic intersections and a new characterization of order-ℓ generic zero patterns, enabling a robust equivalence between higher-order MDS notions. Leveraging the GM-MDS theorem, the authors conclude that generic RS codes are LD-MDS and, in particular, can be list-decoded up to the capacity bound with manageable list sizes, resolving Shangguan–Tamo conjectures in this regime. They further connect these concepts to generic Gabidulin codes, field-size lower bounds, and Maximally Recoverable tensor codes, while offering polynomial-time algorithms to compute generic intersection dimensions via LP duality and invariant theory. These results provide both theoretical unification and practical implications for explicit code design and decoding capabilities at scale.

Abstract

In a recent paper, Brakensiek, Gopi and Makam introduced higher order MDS codes as a generalization of MDS codes. An order-$\ell$ MDS code, denoted by $\operatorname{MDS}(\ell)$, has the property that any $\ell$ subspaces formed from columns of its generator matrix intersect as minimally as possible. An independent work by Roth defined a different notion of higher order MDS codes as those achieving a generalized singleton bound for list-decoding. In this work, we show that these two notions of higher order MDS codes are (nearly) equivalent. We also show that generic Reed-Solomon codes are $\operatorname{MDS}(\ell)$ for all $\ell$, relying crucially on the GM-MDS theorem which shows that generator matrices of generic Reed-Solomon codes achieve any possible zero pattern. As a corollary, this implies that generic Reed-Solomon codes achieve list decoding capacity. More concretely, we show that, with high probability, a random Reed-Solomon code of rate $R$ over an exponentially large field is list decodable from radius $1-R-ε$ with list size at most $\frac{1-R-ε}ε$, resolving a conjecture of Shangguan and Tamo.

Generic Reed-Solomon Codes Achieve List-decoding Capacity

TL;DR

The paper shows that several natural extensions of MDS codes—MDS(ℓ), LD-MDS(≤L), and GZP(ℓ)—are equivalent, up to duality, and that generic Reed-Solomon codes achieve list-decoding capacity for all rates. Central to the results is a dimension formula for generic intersections and a new characterization of order-ℓ generic zero patterns, enabling a robust equivalence between higher-order MDS notions. Leveraging the GM-MDS theorem, the authors conclude that generic RS codes are LD-MDS and, in particular, can be list-decoded up to the capacity bound with manageable list sizes, resolving Shangguan–Tamo conjectures in this regime. They further connect these concepts to generic Gabidulin codes, field-size lower bounds, and Maximally Recoverable tensor codes, while offering polynomial-time algorithms to compute generic intersection dimensions via LP duality and invariant theory. These results provide both theoretical unification and practical implications for explicit code design and decoding capabilities at scale.

Abstract

In a recent paper, Brakensiek, Gopi and Makam introduced higher order MDS codes as a generalization of MDS codes. An order- MDS code, denoted by , has the property that any subspaces formed from columns of its generator matrix intersect as minimally as possible. An independent work by Roth defined a different notion of higher order MDS codes as those achieving a generalized singleton bound for list-decoding. In this work, we show that these two notions of higher order MDS codes are (nearly) equivalent. We also show that generic Reed-Solomon codes are for all , relying crucially on the GM-MDS theorem which shows that generator matrices of generic Reed-Solomon codes achieve any possible zero pattern. As a corollary, this implies that generic Reed-Solomon codes achieve list decoding capacity. More concretely, we show that, with high probability, a random Reed-Solomon code of rate over an exponentially large field is list decodable from radius with list size at most , resolving a conjecture of Shangguan and Tamo.
Paper Structure (43 sections, 38 theorems, 83 equations, 1 table)

This paper contains 43 sections, 38 theorems, 83 equations, 1 table.

Table of Contents

  1. Introduction
  2. List-decoding Reed-Solomon Codes
  3. Previous Work
  4. Higher Order MDS Codes
  5. $\blacktriangleright \operatorname{MDS}(\ell)$.
  6. $\blacktriangleright \operatorname{LD-MDS}(\ell)$.
  7. $\blacktriangleright \operatorname{GZP}(\ell)$.
  8. Equivalence of Higher-order MDS Codes.
  9. Proof Overview
  10. Dimension of generic intersections
  11. A novel characterization of sets in order-$\ell$ generic zero patterns
  12. A generalized Hall's theorem.
  13. Equivalence of $\operatorname{MDS}(\ell+1)$ and $\operatorname{LD-MDS}(\leqslant \ell)^{\perp}$.
  14. Further Applications and Connections
  15. Generic Gabidulin Codes Achieve List-decoding Capacity
  16. ...and 28 more sections

Key Result

theorem 5

Generic Reed-Solomon codes achieve list decoding capacity. If $C$ is a generic $[n,k]$-Reed-Solomon code with rate $R=k/n$, then $C$ is $(\rho,L)$-list decodable for Moreover, $C$ is also $(\rho,L)$-average-radius list-decodable for the same $\rho.$

Theorems & Definitions (79)

  • definition 1
  • definition 2
  • definition 4
  • theorem 5
  • theorem 6
  • remark 7
  • definition 8: $\operatorname{MDS}(\ell)$ bgm2021mds
  • definition 9: $\operatorname{LD-MDS}(L)$ roth2021higher
  • definition 10: Generic zero pattern
  • definition 11: $\operatorname{GZP}(\ell)$
  • ...and 69 more