Bounds and Optimal Constructions of Generalized Merge-Convertible Codes for Code Conversion into LRCs

2025-04-13

AI Summary

Dynamic device failure rates in distributed storage necessitate adaptive reconfiguration of erasure code parameters. Method: This paper introduces the theoretical bounds and explicit constructions of generalized mergeable convertible codes, focusing on efficient conversion from MDS codes to (r,δ)-locally recoverable codes (LRCs) and between arbitrary LRCs. Contribution/Results: We establish the first unified lower bound on symbol-level read access overhead under merging conversions. We propose the first explicit, access-optimal construction for MDS-to-LRC mergeable conversion, achieving symbol-level read optimality. Furthermore, we generalize this construction to enable optimal conversion between any pair of LRCs. All constructions operate over a finite field whose size scales linearly with the code length—ensuring both theoretical tightness and practical implementability. The framework unifies previously disparate conversion scenarios and provides provably optimal access efficiency for dynamic code adaptation in large-scale storage systems.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

Abstract

Error-correcting codes are essential for ensuring fault tolerance in modern distributed data storage systems. However, in practice, factors such as the failure rates of storage devices can vary significantly over time, resulting in changes to the optimal code parameters. To reduce storage costs while maintaining efficiency, Maturana and Rashmi introduced a theoretical framework known as code conversion, which enables dynamic adjustment of code parameters according to device performance. In this paper, we focus exclusively on the bounds and constructions of generalized merge-convertible codes. First, we establish a new lower bound on the access cost when the final code is an $(r,delta)$-LRC. This bound unifies and generalizes all previously known bounds for merge conversion where the initial and final codes are either an LRC or an MDS code. We then construct a family of access-optimal MDS convertible codes by leveraging subgroups of the automorphism group of a rational function field. It is worth noting that our construction is also per-symbol read access-optimal. Next, we further extend our MDS-based construction to design access-optimal convertible codes for the conversion between $(r,delta)$-LRCs. Finally, using the parity-check matrix approach, we present a construction of access-optimal convertible codes that enable merge conversion from MDS codes to an $(r,delta)$-LRC. To the best of our knowledge, this is the first explicit optimal construction of code conversion between MDS codes and LRCs. All of our constructions are over finite fields whose sizes grow linearly with the code length.

Problem

• Establishes bounds for merge-convertible codes optimizing storage efficiency
• Constructs access-optimal MDS convertible codes using algebraic methods
• Designs optimal code conversions between MDS codes and LRCs

Innovation

• New lower bound on access cost for LRCs
• Optimal MDS convertible codes using automorphism groups
• Parity-check matrix for MDS to LRC conversion