Optimal-Access Cooperative MSR Codes: Parity-Check Matrix Construction And a Unified Transformation
Abstract
Cooperative MSR codes are a kind of storage codes which enable optimal-bandwidth repair of any h≥2 node erasures in a cooperative way, while retaining the minimum storage as an [n,k] MDS code. Each code coordinate (node) is assumed to store an array of ℓ symbols, where ℓ is termed as sub-packetization. To address the disk IO (input/output) capability, a cooperative MSR code is said to have optimal-access property, if during node repair, the amount of data accessed at each helper node meets a lower bound on this quantity. In this paper, we focus on reducing the sub-packetization level of optimal-access cooperative MSR codes. We propose new constructions of optimal-access cooperative MSR codes through two methods. At first, we propose a direct explicit construction by designing its parity-check matrix. Such parity-check matrix is built by repeatedly employing two crucial parity-check matrices as building blocks. Secondly, we propose a generic transformation framework. Starting from an arbitrary [n+d−k,d] MDS scalar code, one can derive a final cooperative MSR code by systematically applying two basic transformations. Both approaches yield (n,k,ℓ=δm) optimal-access cooperative MSR codes with δ=d−k+h and m=binomnh−⌊n/δ⌋(binomδh−1). Compared with the state of the art (with ℓ=δbinomnh), the derived codes can reduce the sub-packetization ℓ by a fraction of 1/δ⌊n/δ⌋(binomδh−1), where δ=d−k+h. Moreover, we also show that some previous code structures of optimal-access cooperative MSR codes and optimal-access MSR codes with h=1 are included as special cases of our transformation construction. At last, we note that all of the constructions are built over a finite field of linear size ≥ n+d−k.
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BibTeX
@misc{zhang2026optimalaccess,
title = {{Optimal-Access Cooperative MSR Codes: Parity-Check Matrix Construction And a Unified Transformation}},
author = {Yaqian Zhang and Jingke Xu and Ya-Feng Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06372},
eprint = {2609.06372},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06372},
}
RIS
TY - UNPB
TI - Optimal-Access Cooperative MSR Codes: Parity-Check Matrix Construction And a Unified Transformation
AU - Yaqian Zhang
AU - Jingke Xu
AU - Ya-Feng Liu
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06372
UR - https://arxiv.org/abs/2609.06372
AB - Cooperative MSR codes are a kind of storage codes which enable optimal-bandwidth repair of any h≥2 node erasures in a cooperative way, while retaining the minimum storage as an [n,k] MDS code. Each code coordinate (node) is assumed to store an array of ℓ symbols, where ℓ is termed as sub-packetization. To address the disk IO (input/output) capability, a cooperative MSR code is said to have optimal-access property, if during node repair, the amount of data accessed at each helper node meets a lower bound on this quantity. In this paper, we focus on reducing the sub-packetization level of optimal-access cooperative MSR codes. We propose new constructions of optimal-access cooperative MSR codes through two methods. At first, we propose a direct explicit construction by designing its parity-check matrix. Such parity-check matrix is built by repeatedly employing two crucial parity-check matrices as building blocks. Secondly, we propose a generic transformation framework. Starting from an arbitrary [n+d−k,d] MDS scalar code, one can derive a final cooperative MSR code by systematically applying two basic transformations. Both approaches yield (n,k,ℓ=δ^m) optimal-access cooperative MSR codes with δ=d−k+h and m=binomnh−⌊n/δ⌋(binomδh−1). Compared with the state of the art (with ℓ=δ^binomnh), the derived codes can reduce the sub-packetization ℓ by a fraction of 1/δ^⌊n/δ⌋(binomδh−1), where δ=d−k+h. Moreover, we also show that some previous code structures of optimal-access cooperative MSR codes and optimal-access MSR codes with h=1 are included as special cases of our transformation construction. At last, we note that all of the constructions are built over a finite field of linear size ≥ n+d−k.
ER -