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.

翻译：近期研究表明，分布式存储系统通过根据磁盘故障率动态调整冗余级别可实现显著的空间节约。这种调整通过码转换实现，即初始码编码的数据被转换为最终码编码的数据。尽管该过程通常耗费大量资源，但可转换码旨在高效完成此类转换，同时保持节点修复期间访问的修复度等可译码约束。本文聚焦转换过程中的带宽开销（即传输数据总量），系统研究系统化最优距离局部可修复码（LRC）之间转换的带宽开销理论极限。我们限定研究范围为全局合并模式——在此模式下，多个初始码字合并为单一最终码字，同时保持信息局部性。重点研究稳定可转换码，即最大化转换过程中保持不变节点数量的方案。我们将码转换的信息论建模方法推广至LRC场景，推导出该模式下转换带宽开销的首个非平凡下界。值得注意的是，该下界不依赖于任何线性假设。据此证明，Maturana与Rashmi提出的构造方案在全局合并模式的广泛参数范围内实现了带宽最优性。