  For binary $[n,k,d]$ linear locally repairable codes (LRCs), two new upper
bounds on $k$ are derived. The first one applies to LRCs with disjoint local
repair groups, for general values of $n,d$ and locality $r$, containing some
previously known bounds as special cases. The second one is based on solving an
optimization problem and applies to LRCs with arbitrary structure of local
repair groups. Particularly, an explicit bound is derived from the second bound
when $d\geq 5$. A specific comparison shows this explicit bound outperforms the
Cadambe-Mazumdar bound for $5\leq d\leq 8$ and large values of $n$. Moreover, a
construction of binary linear LRCs with $d\geq6$ attaining our second bound is
provided.
