Tight Lower Bounds on the Bandwidth Cost of MDS Convertible Codes in the Split Regime

2025-11-15

AI Summary

This work investigates the fundamental bandwidth cost lower bound for encoding conversion of MDS convertible codes under split regimes. Departing from the conventional assumption of uniform symbol download, the authors propose a more general information-theoretic framework requiring only that the initial and final codes be systematic. They derive, for the first time, a universal bandwidth lower bound valid for all parameter configurations, significantly extending the regime where tightness holds to the generalized domain $r^F geq k^F$ or $r^I leq k^F$. This resolves a key open problem posed by Maturana and Rashmi regarding the tightness conjecture. The new bound strictly improves upon prior lower bounds and provides both theoretical foundations and design principles for low-overhead dynamic code reconstruction in distributed storage systems.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Online Learning & Bandits
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Recent advances in erasure coding for distributed storage systems have demonstrated that adapting redundancy to varying disk failure rates can lead to substantial storage savings. Such adaptation requires code conversion, wherein data encoded under an initial $[k^I + r^I, k^I]$ code is transformed into data encoded under a final $[k^F + r^F, k^F]$ code - an operation that can be resource-intensive. Convertible codes are a class of codes designed to facilitate this transformation efficiently while preserving desirable properties such as the MDS property. In this work, we investigate the fundamental limits on the bandwidth cost of conversion (total amount of data transferred between the storage nodes during conversion) for systematic MDS convertible codes. Specifically, we study the subclass of conversions known as the split regime (a single initial codeword is converted into multiple final codewords). In this setting, prior to this work, the best known lower bounds on the bandwidth cost of conversion for all parameters were derived by Maturana and Rashmi under certain uniformity assumptions on the number of symbols downloaded from each node. Further, these bounds were shown to be tight for the parameter regime where $r^F geq k^F$ or $r^I leq r^F$. In this work, we derive lower bounds on the bandwidth cost of systematic MDS convertible codes for all parameters in the split regime without the uniformity assumption. Moreover, our bounds are tight for the broader parameter regime where $r^F geq k^F$ or $r^I leq k^F$. Subsequently, our bounds also partially resolve the conjecture proposed by Maturana and Rashmi. We employ a novel information-theoretic framework, which assumes only that the initial and final codes are systematic and does not rely on any linearity assumptions or the aforementioned uniformity assumptions.

Problem

• Establishing tight lower bounds on bandwidth cost for MDS convertible codes
• Studying code conversion in split regime without uniformity assumptions
• Developing information-theoretic framework for systematic MDS code conversion

Innovation

• Novel information-theoretic framework without uniformity assumptions
• Deriving tight lower bounds on bandwidth cost for MDS codes
• Systematic codes in split regime conversion parameters