An Integer Programming-Based Bound for Locally Repairable Codes

Abstract. Locally repairable codes (LRCs) are designed for distributed storage systems to enable efficient repair of failed nodes. This paper derives an upper bound on the dimension of linear LRCs with given length, minimum distance, and locality parameters using an integer programming approach. The bound is obtained by formulating the constraints imposed by the locality structure as integer linear inequalities and solving the resulting optimization problem. The bound improves upon previously known bounds for a wide range of parameters and reveals the optimal trade-off between code parameters. In many cases, the bound is tight, matching known constructions. The integer programming framework provides a unified method for bounding the parameters of LRCs and can be adapted to variants with additional constraints.