Asymptotic Bounds on the Rate of Locally Repairable Codes

New asymptotic upper bounds are presented on the rate of sequences of locally repairable codes (LRCs) with a prescribed relative minimum distance and locality over a finite field
$F$
. The bounds apply to LRCs in which the recovery functions are linear; in particular, the bounds apply to linear LRCs over
$F$
. The new bounds are shown to improve on previously published results, especially when the repair groups are disjoint, namely, they form a partition of the set of coordinates.