Motivated by applications in distributed storage, the notion of a locally recoverable code (LRC) was introduced a few years back. In an LRC, any coordinate of a codeword is recoverable by accessing only a small number of other coordinates. While different properties of LRCs have been well-studied, their performance on channels with random erasures or errors has been mostly unexplored. In this paper, we analyze the performance of LRCs over such stochastic channels. In particular, for input-symmetric discrete memoryless channels, we give a tight characterization of the gap to Shannon capacity when LRCs are used over the channel. Our results hold for a general notion of LRCs that correct multiple local erasures.

翻译：受分布式存储应用的推动，局部可恢复码（LRC）的概念在数年前被提出。在LRC中，码字的任意坐标仅需访问少量其他坐标即可恢复。尽管LRC的多种性质已被充分研究，但其在随机擦除或错误信道上的性能此前鲜有探索。本文分析了LRC在此类随机信道上的表现。具体而言，对于输入对称离散无记忆信道，我们精确刻画了使用LRC时与香农容量之间的差距。该结果适用于可纠正多个局部擦除的广义LRC模型。