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 paper 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
= ⌈(n/r + 1)⌉ and n2 = n
₁
(r +1)-n. Finally, an explicit construction for LRCs attaining this upper bound is presented over the finite field F
_2
m
,where m ≥ n
₁
r. Based on these r ≤ √n - 1 has been definitely determined, which is of great results, the largest possible minimum distance for all LRCs with significance in practical use.