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Lower Bounds on the Sub-Packetization of Optimal-Access MSR Codes for Multiple-Node Repair

Lewen Wang, Zihao Zhang, Sihuang Hu

TL;DR

This paper addresses lower bounds on the sub-packetization level $\ell$ for optimal-access MSR codes in centralized multi-node repair. It develops a linear-algebraic framework for centralized MSR codes, introduces repair matrices and subspaces, and proves rank-based bounds that translate into lower bounds on $\ell$ for both constant repair matrices and matrices independent of helper identities. A novel generating-function approach, parameterized by $c$, $\Delta$, and $\alpha$, yields refined bounds that grow more quickly with $n$ and are especially stronger for high-rate regimes, compared to prior results. The findings inform the feasibility of MSR-code constructions in multi-node failure scenarios and extend to cooperative MSR codes, with practical implications for storage systems design and data-recovery efficiency.

Abstract

We establish lower bounds on the sub-packetization of optimal-access MSR codes in the context of multiple-node failures. These bounds generalize the tight bounds for single-node failure presented by Balaji et al. (IEEE Transactions on Information Theory, vol. 68, no. 10, 2022). Moreover, we utilize generating functions to provide a more refined analysis, further strengthening these bounds.

Lower Bounds on the Sub-Packetization of Optimal-Access MSR Codes for Multiple-Node Repair

TL;DR

This paper addresses lower bounds on the sub-packetization level for optimal-access MSR codes in centralized multi-node repair. It develops a linear-algebraic framework for centralized MSR codes, introduces repair matrices and subspaces, and proves rank-based bounds that translate into lower bounds on for both constant repair matrices and matrices independent of helper identities. A novel generating-function approach, parameterized by , , and , yields refined bounds that grow more quickly with and are especially stronger for high-rate regimes, compared to prior results. The findings inform the feasibility of MSR-code constructions in multi-node failure scenarios and extend to cooperative MSR codes, with practical implications for storage systems design and data-recovery efficiency.

Abstract

We establish lower bounds on the sub-packetization of optimal-access MSR codes in the context of multiple-node failures. These bounds generalize the tight bounds for single-node failure presented by Balaji et al. (IEEE Transactions on Information Theory, vol. 68, no. 10, 2022). Moreover, we utilize generating functions to provide a more refined analysis, further strengthening these bounds.
Paper Structure (14 sections, 12 theorems, 110 equations)

This paper contains 14 sections, 12 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Single-Node Repair
  3. Multiple-Node Repair
  4. Main Results
  5. Outline of This Paper
  6. Linear Algebraic Settings for Centralized Repair
  7. Lower Bounds with Constant Repair Matrices
  8. A Lower Bound with Repair Matrices Independent of Identity of Remaining Helper Nodes
  9. Improving Bounds Using Generating Functions
  10. Generating Functions
  11. A Lower Bound with Constant Repair Matrices
  12. A Lower Bound with Repair Matrices Independent of Identity of Remaining Helper Nodes
  13. Conclusion
  14. The Cut-Set Bound on Centralized MSR Codes With Code Rate No Less Than 1/2

Key Result

Theorem 1

Let $\mathcal{C}$ be an $(n,k,\ell)$ MDS array code. If $\mathcal{C}$ has $(h,d)$ optimal-access repair schemes with constant repair matrices for any $h$ failed systematic nodes, then

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Theorem 4
  • Theorem 5
  • Remark 2
  • Lemma 1
  • ...and 19 more