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A Partial-Exclusion Repair Scheme for MDS Codes

Wei Zhao, Fang-Wei Fu, Ximing Fu

Abstract

For scalar maximum distance separable (MDS) codes, the conventional repair schemes that achieve the cut-set bound with equality for the single-node repair have been proven to require a super-exponential sub-packetization level.As is well known, such an extremely high level severely limits the practical deployment of MDS codes.To address this challenge, we introduce a partial-exclusion (PE) repair scheme for scalar linear codes.In the proposed PE repair framework, each node is associated with an exclusion set.The cardinality of the exclusion set is called the flexibility of the node.The maximum value of flexibility over all nodes defines the \textit{flexibility} of the PE repair scheme. Notably, the conventional repair scheme is the special case of PE repair scheme where the flexibility is 1. Under the PE repair framework, for any valid flexibility, we establish a lower bound on the sub-packetization level of MDS codes that meet the cut-set bound with equality for single-node repair. To realize MDS codes attaining the cut-set bound under the PE repair framework, we propose two generic constructions of Reed-Solomon (RS) codes. Moreover, we demonstrate that for a sufficiently large flexibility, the sub-packetization level of our constructions is strictly lower than the known lower bound established for the conventional repair schemes.This implies that, from the perspective of sub-packetization level, our constructions outperform all existing and potential constructions designed for conventional repair schemes. Finally, we implement the repair process for these codes as executable Magma programs, thereby exhibiting the practical efficiency of our constructions.

A Partial-Exclusion Repair Scheme for MDS Codes

Abstract

For scalar maximum distance separable (MDS) codes, the conventional repair schemes that achieve the cut-set bound with equality for the single-node repair have been proven to require a super-exponential sub-packetization level.As is well known, such an extremely high level severely limits the practical deployment of MDS codes.To address this challenge, we introduce a partial-exclusion (PE) repair scheme for scalar linear codes.In the proposed PE repair framework, each node is associated with an exclusion set.The cardinality of the exclusion set is called the flexibility of the node.The maximum value of flexibility over all nodes defines the \textit{flexibility} of the PE repair scheme. Notably, the conventional repair scheme is the special case of PE repair scheme where the flexibility is 1. Under the PE repair framework, for any valid flexibility, we establish a lower bound on the sub-packetization level of MDS codes that meet the cut-set bound with equality for single-node repair. To realize MDS codes attaining the cut-set bound under the PE repair framework, we propose two generic constructions of Reed-Solomon (RS) codes. Moreover, we demonstrate that for a sufficiently large flexibility, the sub-packetization level of our constructions is strictly lower than the known lower bound established for the conventional repair schemes.This implies that, from the perspective of sub-packetization level, our constructions outperform all existing and potential constructions designed for conventional repair schemes. Finally, we implement the repair process for these codes as executable Magma programs, thereby exhibiting the practical efficiency of our constructions.
Paper Structure (14 sections, 13 theorems, 98 equations, 3 figures, 2 tables)

This paper contains 14 sections, 13 theorems, 98 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Our Results
  3. Organization of the Paper
  4. Preliminaries
  5. Reed-Solomon Codes
  6. Repair schemes for MDS codes
  7. A Partial-Exclusion Linear Repair Scheme
  8. A Trade-Off for PE Repair Schemes
  9. The First Construction of RS Codes with Optimal Repair Bandwidth
  10. The symbol field and the base field
  11. Construction of RS codes
  12. Repair Scheme for a Single Erasure
  13. The Second Construction of RS codes with Optimal Repair Bandwidth
  14. Conclusions

Key Result

Lemma 1

tamo2019repair Let $E$ be an extension of a finite field of $K$. Let $A=(a_{i,j})$ be an $m\times n$ matrix over $E$. Then Moreover, if sfa holds with equality, then for every $\mathcal{J}\subseteq [n]$, where $A_{\mathcal{J}}$ is the restriction of $A$ to the columns with indices in $\mathcal{J}$.

Figures (3)

  • Figure 1: The trade-off between flexibility $t$, the smallest possible sub-packetization level $L$ and the minimum normalized repair bandwidth $\bar{\beta}_{min}$, for the $(14,10)$ RS codes over the base field $\mathbb{F}_2$.
  • Figure 2: The trade-off among the flexibility $t$, the smallest possible sub-packetization level $L$ and the minimum normalized repair bandwidth $\bar{\beta}_{min}$, for $(20,10)$ RS codes over a base field $\mathbb{F}_q$.
  • Figure 3: Inclusion Relationships Among Associated Subfields of the Symbol Field $\mathbb{F}_{2^{2310}}$

Theorems & Definitions (28)

  • Definition 1: A Partial-Exclusion Linear Exact Repair Scheme
  • Remark 1: Repair Locality
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Theorem 5
  • Example 1
  • Remark 3
  • ...and 18 more