  Locally repairable codes (LRCs) are error correcting codes used in
distributed data storage. Besides a global level, they enable errors to be
corrected locally, reducing the need for communication between storage nodes.
There is a close connection between almost affine LRCs and matroid theory which
can be utilized to construct good LRCs and derive bounds on their performance.
  A generalized Singleton bound for linear LRCs with parameters
$(n,k,d,r,\delta)$ was given in [N. Prakash et al., "Optimal Linear Codes with
a Local-Error-Correction Property", IEEE Int. Symp. Inf. Theory]. In this
paper, a LRC achieving this bound is called perfect. Results on the existence
and nonexistence of linear perfect $(n,k,d,r,\delta)$-LRCs were given in [W.
Song et al., "Optimal locally repairable codes", IEEE J. Sel. Areas Comm.].
Using matroid theory, these existence and nonexistence results were later
strengthened in [T. Westerb\"ack et al., "On the Combinatorics of Locally
Repairable Codes", Arxiv: 1501.00153], which also provided a general lower
bound on the maximal achievable minimum distance $d_{\rm{max}}(n,k,r,\delta)$
that a linear LRC with parameters $(n,k,r,\delta)$ can have. This article
expands the class of parameters $(n,k,d,r,\delta)$ for which there exist
perfect linear LRCs and improves the lower bound for
$d_{\rm{max}}(n,k,r,\delta)$. Further, this bound is proved to be optimal for
the class of matroids that is used to derive the existence bounds of linear
LRCs.
