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

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 \(g^I,g^F\le r\).