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$\varepsilon$-MSR Codes for Any Set of Helper Nodes

Vinayak Ramkumar, Netanel Raviv, Itzhak Tamo

TL;DR

This work addresses the practical challenge of repairing failed nodes in distributed storage with near-optimal bandwidth while keeping sub-packetization modest. It introduces a two-building-block framework that first constructs group-algebra–based MDS array codes and then combines them with high-distance codes to yield $\varepsilon$-MSR codes that require no compulsory helper nodes and support any $d$ with $k\le d<n$, achieving $\varepsilon$-optimal repair per helper. The authors extend the approach to MDS array codes capable of repairing $h$ failed nodes with $(h,d)$-epsilon-optimal repair for all feasible $(h,d)$, with sub-packetization scaling logarithmically in code length. These results significantly improve the practicality of MSR-like regenerating codes by reducing sub-packetization while maintaining flexible, distributed repair capabilities, with field sizes scaling nicely with code length.

Abstract

Minimum storage regenerating (MSR) codes are a class of maximum distance separable (MDS) array codes capable of repairing any single failed node by downloading the minimum amount of information from each of the helper nodes. However, MSR codes require large sub-packetization levels, which hinders their usefulness in practical settings. This led to the development of another class of MDS array codes called $\varepsilon$-MSR codes, for which the repair information downloaded from each helper node is at most a factor of $(1+\varepsilon)$ from the minimum amount for some $\varepsilon > 0$. The advantage of $\varepsilon$-MSR codes over MSR codes is their small sub-packetization levels. In previous constructions of epsilon-MSR codes, however, several specific nodes are required to participate in the repair of a failed node, which limits the performance of the code in cases where these nodes are not available. In this work, we present a construction of $\varepsilon$-MSR codes without this restriction. For a code with $n$ nodes, out of which $k$ store uncoded information, and for any number $d$ of helper nodes ($k\le d<n$), the repair of a failed node can be done by contacting any set of $d$ surviving nodes. Our construction utilizes group algebra techniques, and requires linear field size. We also generalize the construction to MDS array codes capable of repairing $h$ failed nodes using $d$ helper nodes with a slightly sub-optimal download from each helper node, for all $h \le r$ and $k \le d \le n-h$ simultaneously.

$\varepsilon$-MSR Codes for Any Set of Helper Nodes

TL;DR

This work addresses the practical challenge of repairing failed nodes in distributed storage with near-optimal bandwidth while keeping sub-packetization modest. It introduces a two-building-block framework that first constructs group-algebra–based MDS array codes and then combines them with high-distance codes to yield -MSR codes that require no compulsory helper nodes and support any with , achieving -optimal repair per helper. The authors extend the approach to MDS array codes capable of repairing failed nodes with -epsilon-optimal repair for all feasible , with sub-packetization scaling logarithmically in code length. These results significantly improve the practicality of MSR-like regenerating codes by reducing sub-packetization while maintaining flexible, distributed repair capabilities, with field sizes scaling nicely with code length.

Abstract

Minimum storage regenerating (MSR) codes are a class of maximum distance separable (MDS) array codes capable of repairing any single failed node by downloading the minimum amount of information from each of the helper nodes. However, MSR codes require large sub-packetization levels, which hinders their usefulness in practical settings. This led to the development of another class of MDS array codes called -MSR codes, for which the repair information downloaded from each helper node is at most a factor of from the minimum amount for some . The advantage of -MSR codes over MSR codes is their small sub-packetization levels. In previous constructions of epsilon-MSR codes, however, several specific nodes are required to participate in the repair of a failed node, which limits the performance of the code in cases where these nodes are not available. In this work, we present a construction of -MSR codes without this restriction. For a code with nodes, out of which store uncoded information, and for any number of helper nodes (), the repair of a failed node can be done by contacting any set of surviving nodes. Our construction utilizes group algebra techniques, and requires linear field size. We also generalize the construction to MDS array codes capable of repairing failed nodes using helper nodes with a slightly sub-optimal download from each helper node, for all and simultaneously.
Paper Structure (20 sections, 9 theorems, 37 equations, 1 figure, 1 table)

This paper contains 20 sections, 9 theorems, 37 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notations and Background
  3. MDS Array Codes
  4. Minimum Storage Regenerating Codes
  5. Small Sub-packetization Level
  6. $\varepsilon$-MSR Codes
  7. Multiple Node Failures and Multiple Repair Degrees
  8. Our Contribution
  9. Construction of $\varepsilon$-MSR Codes
  10. A Group Algebra
  11. An MDS Array Code Construction
  12. Transformation to $\varepsilon$-MSR Codes
  13. Existence of $\varepsilon$-MSR Codes with Logarithmic Sub-packetization Level
  14. Construction of MDS Array Codes with $(h,d)$ $\varepsilon$-optimal repair property property
  15. Repair Property
  16. ...and 5 more sections

Key Result

Theorem 1

For every $a_1,\dots,a_n$, the code constructed in Construction construction:MDS is an $(n,k=n-r,\ell=s^t)_\mathbb{F}$ MDS array code with the following repair property. For any failed node $i\in [n]$ and a subset $D\subseteq [n]\backslash\{i\}$ of $d=k+s-1$ helper nodes, one can repair node $i$ by Moreover, the help-by-transfer property holds.

Figures (1)

  • Figure 1: An illustration of the parity check matrix given by Construction \ref{['construction:eps_MSR']}, for $r=3$, $\lambda=4$ and $t=3$, where $\mathbf{u}^{(1)}=(1,3,2,1)$, $\mathbf{u}^{(2)}=(2,3,1,3)$ and $u^{(3})=(1,2,3,2)$ are three codewords of $\mathcal{U}$. Only the columns corresponding to the three nodes indexed by these three codewords are depicted in the figure.

Theorems & Definitions (16)

  • Example 1
  • Theorem 1
  • proof
  • Remark 2
  • Lemma 4
  • proof
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Proposition 8
  • ...and 6 more