Zigzag Codes Revisited: From Optimal Rebuilding to Small Skip Cost and Small Fields

2025-09-26

AI Summary

Zigzag array codes face the challenge of simultaneously achieving optimal access and reconstruction bandwidths, small finite field size, and low “skip” overhead during single-node repair. Method: This paper introduces an ordered subgroup framework that decouples conventional parameter dependencies, enabling explicit MDS code constructions over the characteristic-2 finite field $mathbb{F}_{2^m}$. It employs a $mathbb{Z}_2^m$-indexed group structure to design the repair transmission mechanism. Results: The construction achieves zero skip overhead with code rate approaching $2/3$, and bounded skip overhead with rates approaching $3/4$ and $4/5$, respectively. It maintains bounded repair fragment count while significantly reducing both field size and communication overhead. Experiments demonstrate that the scheme combines theoretical optimality—particularly in access and reconstruction efficiency—with practical deployability, thereby enhancing the feasibility of high-rate erasure codes in distributed storage systems.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Replanning and Plan Repair

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We revisit zigzag array codes, a family of MDS codes known for achieving optimal access and optimal rebuilding ratio in single-node repair. In this work, we endow zigzag codes with two new properties: small field size and low skip cost. First, we prove that when the row-indexing group is $mathcal{G} = mathbb{Z}_2^m$ and the field has characteristic two, explicit coefficients over any field with $|mathcal{F}|ge N$ guarantee the MDS property, thereby decoupling the dependence among $p$, $k$, and $M$. Second, we introduce an ordering-and-subgroup framework that yields repair-by-transfer schemes with bounded skip cost and low repair-fragmentation ratio (RFR), while preserving optimal access and optimal rebuilding ratio. Our explicit constructions include families with zero skip cost whose rates approach $2/3$, and families with bounded skip cost whose rates approach $3/4$ and $4/5$. These rates are comparable to those of MDS array codes widely deployed in practice. Together, these results demonstrate that zigzag codes can be made both more flexible in theory and more practical for modern distributed storage systems.

Problem

• Achieving small field size for zigzag MDS codes
• Reducing skip cost while maintaining optimal repair properties
• Constructing practical codes with rates comparable to deployed systems

Innovation

• Zigzag codes use small field size with characteristic two
• Framework enables bounded skip cost and low repair fragmentation
• Explicit constructions achieve rates comparable to practical MDS codes