Bounds and Constructions for Linear Locally Repairable Codes over Binary Fields

Abstract. We investigate linear locally repairable codes (LRCs) over binary fields, deriving new upper bounds on the code dimension for given locality and distance parameters. Our bounds are tighter than existing general bounds for the binary case, reflecting the constraints imposed by the binary field. We also present explicit constructions of binary LRCs that achieve or approach the derived bounds, demonstrating the optimality of our results. The constructions are based on combinatorial designs and algebraic methods tailored to binary alphabets. Our work contributes to the understanding of the fundamental limits of LRCs over small fields, which are relevant for practical distributed storage systems.