Locally repairable codes (LRCs) were originally introduced to enable efficient recovery from erasures in distributed storage systems by accessing only a small number of other symbols. While their structural properties-such as bounds and constructions-have been extensively studied, the performance of LRCs under random erasures and errors has remained largely unexplored. In this work, we study the error- and erasure-correction performance of binary linear LRCs under majority-logic decoding (MLD). Focusing on LRCs with fixed locality and varying availability, we derive explicit upper bounds on the probability of decoding failure over the memoryless Binary Erasure Channel (BEC) and Binary Symmetric Channel (BSC). Our analysis characterizes the behavior of the bit-error rate (BER) and block-error rate (BLER) as functions of the locality and availability parameters. We show that, under mild growth conditions on the availability, the block decoding failure probability vanishes asymptotically, and that majority-logic decoding can successfully correct virtually all of error and erasure patterns of weight linear in the blocklength. The results reveal a substantial gap between worst-case guarantees and typical performance under stochastic channel models.

翻译：局部可修复码最初被引入，旨在通过仅访问少量其他符号，实现分布式存储系统中擦除的高效恢复。尽管其结构特性——如界与构造——已得到广泛研究，但LRC在随机擦除与错误下的性能在很大程度上仍未得到探索。本文研究了二进制线性LRC在择多逻辑译码下的纠错与擦除恢复性能。针对具有固定局部性及可变可用性的LRC，我们推导了在无记忆二进制擦除信道与二进制对称信道上译码失败概率的显式上界。我们的分析刻画了误比特率与误块率作为局部性和可用性参数的函数行为。研究表明，在可用性满足温和增长条件下，块译码失败概率渐近趋于零，且择多逻辑译码能够成功纠正几乎所有权重与码长呈线性关系的错误与擦除模式。这些结果揭示了最坏情况保证与随机信道模型下的典型性能之间存在显著差距。