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An Integer Programming Based Bound for Locally Repairable Codes

The locally repairable code (LRC) studied in this paper is an $[n,k]$ linear code of which the value at each coordinate can be recovered by a linear combination of at most $r$ other coordinates. The central problem in this work is to determine the largest possible minimum distance for LRCs. First, an integer programming based upper bound is derived for any LRC. Then by solving the programming problem under certain conditions, an explicit upper bound is obtained for LRCs with parameters $n_1>n_2$, where $n_1 = \left\lceil \frac{n}{r+1} \right\rceil$ and $n_2 = n_1 (r+1) - n$. Finally, an explicit construction for LRCs attaining this upper bound is presented over the finite field $\mathbb{F}_{2^m}$, where $m\geq n_1r$. Based on these results, the largest possible minimum distance for all LRCs with $r \le \sqrt{n}-1$ has been definitely determined, which is of great significance in practical use.