Recent studies have shown that distributed storage systems can achieve significant space savings by adapting redundancy levels to varying disk failure rates. This adaptation is performed via code conversion, wherein data encoded under an initial code are transformed to data encoded under a final code. While this process is typically resource-intensive, convertible codes are designed to enable these transformations efficiently while preserving desirable decodability constraints such as repair degree, or the number of nodes accessed during node repair. In this work, we focus on the bandwidth cost of conversion, or the total amount of data transferred during the conversion process. We study fundamental limits on the bandwidth cost of conversion between systematic optimal-distance Locally Repairable Codes (LRCs). We restrict our focus to the global merge regime, in which multiple initial codewords are combined to form a single final codeword while preserving information locality. We focus on stable convertible codes, wherein the number of unchanged nodes is maximized during conversion. We generalize an information-theoretic approach for modeling code conversion to the LRC setting, and derive the first non-trivial lower bounds on the bandwidth cost of conversion in this regime. Notably, our bounds do not rely on any linearity assumptions. Consequently, we show that the constructions of Maturana and Rashmi are bandwidth-optimal across a broad range of parameters in the global merge regime.

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