pith. sign in

arxiv: 2605.03991 · v1 · submitted 2026-05-05 · 💻 cs.IT · math.IT

Joint Design of Piggyback and Conjugate Transformation Functions for Repair Bandwidth Reduction in Piggybacking Codes

Hao Shi , Zhengyi Jiang , Gefeng Deng , Zhongyi Huang , Hanxu Hou This is my paper

Pith reviewed 2026-05-07 13:30 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords piggybacking codesconjugate transformationsrepair bandwidthMDS array codesdistributed storageerasure codingrepair efficiency
0
0 comments X

The pith

Conjugate-piggybacking codes reduce repair bandwidth by jointly designing piggyback functions and conjugate transformations while preserving MDS over moderate fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces conjugate-piggybacking codes that combine piggyback functions with conjugate transformations for MDS array codes. The joint design maintains the minimum distance property over fields such as F_{2^8} without raising sub-packetization levels. It allows certain parity nodes to repair with optimal bandwidth and lowers the total repair bandwidth relative to prior piggybacking approaches. Simulations confirm reduced expected repair traffic under single-node failures compared to standard Reed-Solomon repair, though at the expense of a modestly larger field.

Core claim

The proposed conjugate-piggybacking codes achieve improved repair efficiency in high-rate distributed storage by jointly optimizing piggyback functions and conjugate transformations. This construction ensures the array code remains MDS over moderate field sizes like F_{2^8} and enables optimal repair bandwidth for some parity nodes, resulting in lower overall repair bandwidth than existing piggybacking designs.

What carries the argument

The conjugate-piggybacking construction, which integrates conjugate transformations into the design of piggyback functions to enhance repair efficiency in MDS array codes.

If this is right

  • Some parity nodes achieve optimal repair bandwidth.
  • The overall repair bandwidth is reduced compared to existing piggybacking-based designs.
  • The MDS property is preserved over moderate field sizes such as F_{2^8}.
  • Expected repair traffic is lower than conventional RS repair in simulations of uniform single-node failures.
  • These gains are obtained without increasing the sub-packetization level.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This method might apply to other types of erasure codes used in distributed systems.
  • Practical implementations could lead to lower network costs in large-scale storage clusters.
  • Further research could explore extensions to multiple node failures or dynamic repair scenarios.
  • The trade-off with field size suggests potential for optimization in hardware with larger field support.

Load-bearing premise

The piggyback functions and conjugate transformations can be jointly selected to ensure the code stays MDS over moderate fields without raising sub-packetization.

What would settle it

A counterexample would be a specific joint design over F_{2^8} that either violates the MDS property for some data patterns or fails to show bandwidth reduction in repair simulations.

Figures

Figures reproduced from arXiv: 2605.03991 by Gefeng Deng, Hanxu Hou, Hao Shi, Zhengyi Jiang, Zhongyi Huang.

Figure 1
Figure 1. Figure 1: An example of (n, k, ℓ) = (14, 10, 4) conjugate-piggybacking codes. The array G1 consists of four instances of a (14, 10, 1) systematic MDS code. The array G2 is obtained after the piggybacking step, and the array G3 is obtained after the transformation step. The red symbols in G3 are introduced by the transformation step. bandwidth of data nodes, while the conjugate transformation drives some parity nodes… view at source ↗
Figure 2
Figure 2. Figure 2: The structure of G2, where the blue part represents the region R, the red part represents the region M1, and the yellow part represents the region M2. B. MDS Property We next establish a sufficient condition under which the proposed codes are MDS. Theorem 1. If the field size satisfies q > kr2 , then the codes C(n, k, L) are MDS codes. Proof. It suffices to show that C(n, k, L) can recover from any r = n −… view at source ↗
Figure 3
Figure 3. Figure 3: The triangle area Q, i.e., the orange part (△ABC). Since Q is a bounded closed domain and γ8(m, L) is a continuous function of two variables on Q, the minimum value is attained. We next show by contradiction that the minimum of γ8(m, L) cannot be attained at an interior point of Q. Suppose that γ8(m, L) attains the minimum at an interior point (m′ , L′ ) of Q. Then ∂γ8(m′ , L′ ) ∂m = 0 ⇒ 1 L′(r − 1) − ( L … view at source ↗
Figure 4
Figure 4. Figure 4: Average repair bandwidth ratio of all nodes for the proposed code, view at source ↗
read the original abstract

Efficient node repair is a central requirement in distributed storage systems, particularly in high-rate erasure-coded deployments where repair traffic directly affects network overhead and recovery cost. Piggybacking codes reduce the repair bandwidth of MDS array codes while keeping the sub-packetization level small. However, existing piggybacking constructions often rely on restrictive piggyback-function designs to preserve the MDS property over small fields, which limits their repair-bandwidth reduction. We propose {\em conjugate-piggybacking} codes, a new class of MDS array codes that jointly design piggyback functions and conjugate transformations under small sub-packetization. The proposed construction improves repair efficiency while preserving the MDS property over moderate field sizes. In particular, it enables some parity nodes to achieve optimal repair bandwidth and reduces the overall repair bandwidth compared with existing piggybacking-based designs. We analyze the MDS property and repair bandwidth of the proposed codes and evaluate them against existing piggybacking codes under high-code-rate settings over $\mathbb{F}_{2^8}$. We further conduct a repair-traffic simulation under uniform single-node failures to quantify the expected traffic reduction in storage-oriented settings. The results show that our construction consistently achieves lower repair bandwidth than related piggybacking codes and reduces expected repair traffic compared with conventional RS repair. These gains are obtained at the cost of a slightly larger field size, revealing a practical trade-off between repair efficiency and field-size overhead for high-rate distributed storage.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes conjugate-piggybacking codes, a new class of MDS array codes obtained by jointly designing piggyback functions and conjugate transformations. The construction is claimed to preserve the MDS property over moderate fields such as F_{2^8} while reducing repair bandwidth relative to prior piggybacking schemes, enabling optimal repair for some parity nodes, and lowering expected repair traffic under single-node failures as shown by analysis and simulations.

Significance. If the MDS guarantee and bandwidth claims hold, the work is significant for high-rate distributed storage, where repair overhead is a key cost. The joint-design approach and the explicit repair-traffic simulation under uniform failures provide practical value beyond pure bandwidth expressions. The acknowledged field-size trade-off is useful for system-level assessment.

major comments (2)
  1. [Section IV] Section IV (MDS Property Analysis): the argument that the conjugate map preserves full rank for every k-column submatrix over F_{2^8} must be strengthened with an explicit rank calculation or lemma that rules out linear dependence for the chosen piggyback functions, especially at high rates (k close to n). The current analysis appears to rely on generic position without addressing the multiplicative structure of the conjugate transformation, which directly affects the central MDS claim.
  2. [Section V] Section V (Repair Bandwidth): the statement that certain parity nodes achieve optimal repair bandwidth is load-bearing for the improvement claim, yet it is not shown to hold independently of the specific function choices or for all data-symbol placements; a parameter-independent derivation or counter-example check is needed.
minor comments (2)
  1. [Abstract] The abstract and introduction should quantify the exact field-size increase (e.g., from F_{2^4} to F_{2^8}) and the sub-packetization level relative to the baselines being compared.
  2. [Simulation section] Simulation figures would benefit from inclusion of the information-theoretic repair-bandwidth lower bound as a reference line and from reporting the number of random trials or confidence intervals.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our paper. We address the major comments point by point below, and will revise the manuscript accordingly to strengthen the relevant sections.

read point-by-point responses

  1. Referee: Section IV (MDS Property Analysis): the argument that the conjugate map preserves full rank for every k-column submatrix over F_{2^8} must be strengthened with an explicit rank calculation or lemma that rules out linear dependence for the chosen piggyback functions, especially at high rates (k close to n). The current analysis appears to rely on generic position without addressing the multiplicative structure of the conjugate transformation, which directly affects the central MDS claim.

    Authors: We agree that an explicit rank calculation would strengthen the MDS property analysis. The original argument leverages the conjugate transformation's properties to maintain full rank, but to directly address the multiplicative structure, we will add a dedicated lemma in the revised Section IV. This lemma will explicitly compute the rank of k-column submatrices over F_{2^8}, showing that the chosen piggyback functions prevent linear dependence by exploiting the field automorphism. We will include a proof sketch that covers high-rate regimes (k close to n) to ensure the MDS property holds rigorously. revision: yes

  2. Referee: Section V (Repair Bandwidth): the statement that certain parity nodes achieve optimal repair bandwidth is load-bearing for the improvement claim, yet it is not shown to hold independently of the specific function choices or for all data-symbol placements; a parameter-independent derivation or counter-example check is needed.

    Authors: The optimal repair bandwidth for certain parity nodes follows from the joint optimization in our construction. To make this independent of specific choices, we will provide a parameter-independent derivation in the revised Section V, demonstrating that as long as the piggyback functions satisfy the MDS condition with the conjugate map, the repair bandwidth for those nodes reaches the optimal value. Additionally, we will verify this for various data-symbol placements through a general argument rather than specific examples, ensuring no counter-examples exist under the proposed design. revision: yes

Circularity Check

0 steps flagged

No circularity: conjugate-piggybacking construction is an independent algebraic design with explicit MDS and bandwidth analysis

full rationale

The paper defines a new joint piggyback-conjugate transformation explicitly, then separately proves the resulting array code remains MDS over F_{2^8} and computes its repair bandwidth. No equation or claim reduces the MDS guarantee or bandwidth savings to a fitted parameter, a self-citation chain, or a renaming of prior results; the construction parameters are chosen by the authors and the properties are verified independently of the performance claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, new entities, or non-standard axioms are described beyond the usual assumptions of linear algebra over finite fields and the MDS property definition.

pith-pipeline@v0.9.0 · 5571 in / 1256 out tokens · 76775 ms · 2026-05-07T13:30:05.086712+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

25 extracted references · 2 canonical work pages

  1. [1]

    Conjugate- Piggybacking Codes: MDS Array Codes with Lower Repair Bandwidth over Small Field Size,

    H. Shi, Z. Jiang, Z. Huang, B. Bai, G. Zhang, and H. Hou, “Conjugate- Piggybacking Codes: MDS Array Codes with Lower Repair Bandwidth over Small Field Size,” in2024 IEEE Information Theory Workshop (ITW) (ITW’2024), Shenzhen, China, November 2024, p. 5.49

    2024

  2. [2]

    Polynomial Codes over Certain Finite Fields,

    I. S. Reed and G. Solomon, “Polynomial Codes over Certain Finite Fields,”Journal of the Society for Industrial & Applied Mathematics, vol. 8, no. 2, pp. 300–304, 1960

    1960

  3. [3]

    Network Coding for Distributed Storage Systems,

    A. Dimakis, P. Godfrey, Y . Wu, M. Wainwright, and K. Ramchandran, “Network Coding for Distributed Storage Systems,”IEEE Trans. Infor- mation Theory, vol. 56, no. 9, pp. 4539–4551, Sep. 2010

    2010

  4. [4]

    Optimal Exact-Regenerating Codes for Distributed Storage at the MSR and MBR Points via a Product- Matrix Construction,

    K. V . Rashmi, N. B. Shah, and P. V . Kumar, “Optimal Exact-Regenerating Codes for Distributed Storage at the MSR and MBR Points via a Product- Matrix Construction,”IEEE Trans. Information Theory, vol. 57, no. 8, pp. 5227–5239, August 2011

    2011

  5. [5]

    Zigzag Codes: MDS Array Codes with Optimal Rebuilding,

    I. Tamo, Z. Wang, and J. Bruck, “Zigzag Codes: MDS Array Codes with Optimal Rebuilding,”IEEE Trans. Information Theory, vol. 59, no. 3, pp. 1597–1616, May 2013

    2013

  6. [6]

    BASIC Codes: Low- Complexity Regenerating Codes for Distributed Storage Systems,

    H. Hou, K. W. Shum, M. Chen, and H. Li, “BASIC Codes: Low- Complexity Regenerating Codes for Distributed Storage Systems,”IEEE Trans. Information Theory, vol. 62, no. 6, pp. 3053–3069, 2016

    2016

  7. [7]

    Explicit Constructions of Optimal-Access MDS Codes with Nearly Optimal Sub-Packetization,

    M. Ye and A. Barg, “Explicit Constructions of Optimal-Access MDS Codes with Nearly Optimal Sub-Packetization,”IEEE Transactions on Information Theory, vol. 63, no. 10, pp. 6307–6317, 2017

    2017

  8. [8]

    A Generic Transformation to Enable Optimal Repair in MDS codes for Distributed Storage Systems,

    J. Li, X. Tang, and C. Tian, “A Generic Transformation to Enable Optimal Repair in MDS codes for Distributed Storage Systems,”IEEE Trans. Information Theory, vol. 64, no. 9, pp. 6257–6267, 2018

    2018

  9. [9]

    A Tight Lower Bound on the Sub- Packetization Level of Optimal-Access MSR and MDS Codes,

    S. B. Balaji and P. V . Kumar, “A Tight Lower Bound on the Sub- Packetization Level of Optimal-Access MSR and MDS Codes,” inProc. IEEE Int. Symp. Inf. Theory, 2018, pp. 2381–2385

    2018

  10. [10]

    Multi-Layer Transformed MDS Codes with Optimal Repair Access and Low Sub-Packetization,

    H. Hou, P. P. C. Lee, and Y . S. Han, “Multi-Layer Transformed MDS Codes with Optimal Repair Access and Low Sub-Packetization,”arXiv preprint arXiv:1907.08938, 2019

    work page arXiv 1907

  11. [11]

    Binary MDS Array Codes with Optimal Repair,

    H. Hou and P. P. C. Lee, “Binary MDS Array Codes with Optimal Repair,” IEEE Trans. Information Theory, vol. 66, no. 3, pp. 1405—-1422, Mar. 2020

    2020

  12. [12]

    A Piggybacking Design Framework for Read-and Download-efficient Distributed Storage Codes,

    K. V . Rashmi, N. B. Shah, and K. Ramchandran, “A Piggybacking Design Framework for Read-and Download-efficient Distributed Storage Codes,”IEEE Trans. Information Theory, vol. 63, no. 9, pp. 5802–5820, 2017

    2017

  13. [13]

    A Repair-Efficient Coding for Distributed Storage Systems Under Piggybacking Framework,

    S. Yuan, Q. Huang, and Z. Wang, “A Repair-Efficient Coding for Distributed Storage Systems Under Piggybacking Framework,”IEEE Trans. Communications, vol. 66, no. 8, pp. 3245–3254, 2018

    2018

  14. [14]

    A New Piggybacking Design with Low Repair Bandwidth and Complexity,

    R. Sun, L. Zhang, and J. Liu, “A New Piggybacking Design with Low Repair Bandwidth and Complexity,”IEEE Communications Letters, vol. 25, no. 7, pp. 2099–2103, 2021

    2099

  15. [15]

    An efficient piggybacking design with lower repair bandwidth and lower sub-packetization,

    Z. Jiang, H. Hou, Y . S. Han, Z. Huang, B. Bai, and G. Zhang, “An efficient piggybacking design with lower repair bandwidth and lower sub-packetization,” inProc. IEEE Int. Symp. Inf. Theory, 2021, pp. 2328– 2333

    2021

  16. [16]

    An Efficient One-to-One Piggybacking Design for Distributed Storage Systems,

    G. Y . Li, X. Lin, and X. Tang, “An Efficient One-to-One Piggybacking Design for Distributed Storage Systems,”IEEE Trans. Communications, vol. 67, no. 12, pp. 8193–8205, 2019

    2019

  17. [17]

    New Piggybacking Codes with Lower Repair Bandwidth for Any Single- Node Failure,

    H. Shi, H. Hou, Y . S. Han, P. P. C. Lee, Z. Jiang, Z. Huang, and B. Bai, “New Piggybacking Codes with Lower Repair Bandwidth for Any Single- Node Failure,” in2022 IEEE International Symposium on Information Theory (ISIT), 2022, pp. 2601–2606

    2022

  18. [18]

    Toward Lower Repair Bandwidth of Piggybacking Codes via Jointly Design for Both Data and Parity Nodes,

    Z. Jiang, H. Shi, Z. Huang, B. Bai, G. Zhang, and H. Hou, “Toward Lower Repair Bandwidth of Piggybacking Codes via Jointly Design for Both Data and Parity Nodes,” inGLOBECOM 2023 - 2023 IEEE Global Communications Conference, 2023, pp. 7345–7350

    2023

  19. [19]

    Toward Lower Repair Bandwidth and Optimal Repair Complexity of Piggybacking Codes with Small Sub-packetization,

    ——, “Toward Lower Repair Bandwidth and Optimal Repair Complexity of Piggybacking Codes with Small Sub-packetization,”IEEE Transactions on Communications, 2024

    2024

  20. [20]

    HashTag Erasure Codes: From Theory to Practice,

    K. Kralevska, D. Gligoroski, R. E. Jensen, and H. Øverby, “HashTag Erasure Codes: From Theory to Practice,”IEEE Transactions on Big Data, vol. 4, no. 4, pp. 516–529, 2018

    2018

  21. [21]

    Bidirectional Piggybacking Design for All Nodes With Sub-Packetization 2≤l≤r ,

    K. Wang and Z. Zhang, “Bidirectional Piggybacking Design for All Nodes With Sub-Packetization 2≤l≤r ,”IEEE Transactions on Communications, vol. 71, no. 12, pp. 6859–6869, 2023

    2023

  22. [22]

    Bidirectional Piggybacking Design for All Nodes with Sub- Packetization l = r,

    ——, “Bidirectional Piggybacking Design for All Nodes with Sub- Packetization l = r,” in2023 IEEE Information Theory Workshop (ITW), 2023, pp. 305–310

    2023

  23. [23]

    Balancing Repair Bandwidth and Sub-Packetization in Erasure-Coded Storage via Elastic Transformation,

    K. Tang, K. Cheng, H. H. W. Chan, X. Li, P. P. C. Lee, Y . Hu, J. Li, and T.-Y . Wu, “Balancing Repair Bandwidth and Sub-Packetization in Erasure-Coded Storage via Elastic Transformation,” inIEEE INFOCOM 2023 - IEEE Conference on Computer Communications, 2023, pp. 1–10

    2023

  24. [24]

    Piggybacking+ Codes: MDS Array Codes with Linear Sub-Packetization to Achieve Lower Repair Bandwidth,

    H. Shi, Z. Jiang, Z. Huang, B. Bai, G. Zhang, and H. Hou, “Piggybacking+ Codes: MDS Array Codes with Linear Sub-Packetization to Achieve Lower Repair Bandwidth,” inGLOBECOM 2023 - 2023 IEEE Global Communications Conference, 2023, pp. 7351–7356

    2023

  25. [25]

    Two Piggybacking Codes with Flexible Sub-Packetization to Achieve Lower Repair Bandwidth,

    H. Shi, Z. Jiang, Z. Huang, B. Bai, and H. Hou, “Two Piggybacking Codes with Flexible Sub-Packetization to Achieve Lower Repair Bandwidth,” p. arXiv:2209.09691, September 2022

    work page arXiv 2022