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Maximal Recoverability: A Nexus of Coding Theory

Joshua Brakensiek, Venkatesan Guruswami

TL;DR

This survey dives into the study of two families of MR codes: MR locally recoverable codes (LRCs) and grid codes (GCs), and discusses the primary recoverability guarantees as well as what is known concerning optimal constructions.

Abstract

In the modern era of large-scale computing systems, a crucial use of error correcting codes is to judiciously introduce redundancy to ensure recoverability from failure. To get the most out of every byte, practitioners and theorists have introduced the framework of maximal recoverability (MR) to study optimal error-correcting codes in various architectures. In this survey, we dive into the study of two families of MR codes: MR locally recoverable codes (LRCs) (also known as partial MDS codes) and grid codes (GCs). For each of these two families of codes, we discuss the primary recoverability guarantees as well as what is known concerning optimal constructions. Along the way, we discuss many surprising connections between MR codes and broader questions in computer science and mathematics. For MR LRCs, the use of skew polynomial codes has unified many previous constructions. For MR GCs, the theory of higher order MDS codes shows that MR GCs can be used to construct optimal list-decodable codes. Furthermore, the optimally recoverable patterns of MR GCs have close ties to long-standing problems on the structural rigidity of graphs.

Maximal Recoverability: A Nexus of Coding Theory

TL;DR

This survey dives into the study of two families of MR codes: MR locally recoverable codes (LRCs) and grid codes (GCs), and discusses the primary recoverability guarantees as well as what is known concerning optimal constructions.

Abstract

In the modern era of large-scale computing systems, a crucial use of error correcting codes is to judiciously introduce redundancy to ensure recoverability from failure. To get the most out of every byte, practitioners and theorists have introduced the framework of maximal recoverability (MR) to study optimal error-correcting codes in various architectures. In this survey, we dive into the study of two families of MR codes: MR locally recoverable codes (LRCs) (also known as partial MDS codes) and grid codes (GCs). For each of these two families of codes, we discuss the primary recoverability guarantees as well as what is known concerning optimal constructions. Along the way, we discuss many surprising connections between MR codes and broader questions in computer science and mathematics. For MR LRCs, the use of skew polynomial codes has unified many previous constructions. For MR GCs, the theory of higher order MDS codes shows that MR GCs can be used to construct optimal list-decodable codes. Furthermore, the optimally recoverable patterns of MR GCs have close ties to long-standing problems on the structural rigidity of graphs.
Paper Structure (27 sections, 14 theorems, 8 equations, 2 figures)

This paper contains 27 sections, 14 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximally Recoverable Locally Recoverable Codes
  4. Maximally Recoverable Grid and Tensor Codes
  5. Characterizations of Correctable Patterns
  6. Case Study: a=b=1
  7. The a=1 Case
  8. The a >= 2 Case
  9. "If" direction.
  10. "Only if" direction.
  11. Field Size Bounds and Explicit Constructions
  12. MR Tensor Codes
  13. MR Grid Codes
  14. Higher Order MDS Codes
  15. Connections to List-decoding
  16. ...and 12 more sections

Key Result

Proposition 2

Let $H$ be the parity check matrix of a $[n,k]$-code $C$. We have that $E \subseteq [n]$ is recoverable if and only if the columns of $H$ spanned by $E$ are linearly independent--that is, $\operatorname{rank} H|_{E} = |E|$. In particular, when $|E| =n-k$, $E$ is recoverable iff $\det (H|_{E}) \neq 0

Figures (2)

  • Figure 1: This diagram visualizes the encoding map of a locally recoverable code. First, $k$ data symbols are expanded to $k+h$ symbols using $h$ global parity checks. Then, these symbols are broken up into $\frac{k+h}{r-a}$ local groups, each of which is expanded with $a$ more parity checks. Illustration adapted from Figure 1 of gopi2020maximally.
  • Figure 2: An illustration of a pattern which is not correctable in a $(5,5,2,2)$-MR tensor code as discovered by Holzbauer et al. holzbaur2021correctable. Each represents an erased symbol.

Theorems & Definitions (21)

  • Definition 1: MR
  • Proposition 2
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5: holzbaur2021correctable
  • Theorem 6: e.g., Gopalan2016
  • Definition 7
  • Theorem 8: Gopalan2016
  • Theorem 9: bernstein2017completion as stated in brakensiek2024rigidity
  • ...and 11 more