Tight Lower Bounds and Optimal Constructions of Locally Repairable Convertible Codes in the Split Regime

2026-06-25

AI Summary

This work addresses the problem of minimizing read bandwidth during parameter adaptation of locally repairable codes (LRCs) in distributed storage systems. Focusing on systematic maximum-distance-separable (MDS) LRCs under global splitting scenarios where both the initial and final numbers of global parity nodes satisfy \(g^I, g^F \leq r\) (with \(r\) denoting the local dimension), the study derives, for the first time, a tight information-theoretic lower bound on read bandwidth without assuming linearity. Building upon MDS array codes with prescribed repair properties, the paper proposes three explicit constructions that achieve this bound for all cases: \(g^F = g^I\), \(g^F > g^I\), and \(g^F < g^I\). These schemes thus attain optimal conversion cost in the specified parameter regime.

Abstract

Erasure codes are a key technique for achieving fault-tolerant storage in modern distributed storage systems. As storage systems evolve, their code parameters often need to be adjusted to accommodate changes in storage scale, reliability requirements, and disk failure rates. Such adaptation is performed through code conversion, where data encoded under an initial code are transformed into data encoded under a final code. Convertible codes are designed to carry out this transformation efficiently while preserving desirable properties of the underlying codes. In this work, we study conversions between systematic optimal-distance locally repairable codes (LRCs), focusing on the read-bandwidth cost in the global split regime. We concentrate on the parameter range \(g^I,g^F\le r\), where the numbers of initial and final global parity nodes do not exceed the local dimension. For this entire range, we derive read-bandwidth lower bounds for stable optimal-distance locally repairable convertible codes (LRCCs) via an information-theoretic approach, without imposing any linearity assumption on the initial and final codes or on the conversion procedure. We then develop constructions based on MDS array codes with prescribed repair properties. According to the relative sizes of \(g^I\) and \(g^F\), the construction is divided into the three cases \(g^F=g^I\), \(g^F>g^I\), and \(g^F<g^I\); in each case, it attains the corresponding lower bound. Hence we characterize the optimal read-bandwidth cost for stable optimal-distance LRCCs throughout the parameter range \(g^I,g^F\le r\).

Problem

• locally repairable codes
• convertible codes
• read-bandwidth
• code conversion
• distributed storage

Innovation

• locally repairable convertible codes
• read-bandwidth lower bounds
• optimal-distance LRCs
• MDS array codes
• split regime