Bounds for Binary Linear Locally Repairable Codes via a Sphere-Packing Approach

Abstract. Locally repairable codes (LRCs) are a class of erasure codes designed for distributed storage systems where a failed node can be repaired by accessing only a small number of other nodes. For binary linear LRCs, we derive new upper bounds on the code dimension by employing a sphere-packing approach that accounts for the locality structure. Our bounds improve upon previously known bounds for a wide range of parameters, providing tighter characterizations of the trade-off between locality, distance, and dimension. We also present constructions that achieve or approach these bounds, demonstrating the tightness of our results. The sphere-packing framework introduced in this work offers a systematic method for bounding the parameters of LRCs and can be extended to more general settings.