Optimal Binary Locally Repairable Codes with Locality and Availability from Latin Squares

The rapid development of machine learning, large language models, and related technologies has greatly increased the demand for data storage capacity. Therefore, the role of distributed storage systems in such applications has become more prominent. However, it is inevitable that a single node fails in a distributed storage system during long-term use. Being able to repair failed nodes in a timely manner is extremely important for the stable operation of distributed storage systems, and a specific encoding scheme is required to meet the needs of efficiently repairing failed nodes. This research presents a novel family of binary locally repairable codes (LRCs) developed using multiple disjoint recovery sets constructed based on mutually orthogonal Latin squares (MOLS). The proposed constructions achieve distance optimality under the Singleton-like bound for LRCs with availability. Specifically, the codes are parameterized as (n=r2+tr,k=r2,r,t) and (n=rm+tm,k=rm,r,t), where n is the block length, k is the dimension, r is the locality, and t is the availability. These codes achieve minimum distance d=t+1, guaranteeing efficient recovery with t disjoint repair sets, each of size r. Compared to existing constructions, the proposed codes offer significant improvements in terms of code rate R=rr+t, support for larger block lengths, and reduced finite field size requirements (field size q=2). Additionally, a method is introduced to improve the minimum distance of codes with even availability t, constructing codes with parameters (n+1,k,r,t) and d=t+2, while preserving optimality. These properties make the proposed codes particularly suitable for distributed storage systems, where efficient and parallel repair of failed nodes is critical.