  Locally repairable codes (LRCs) with $(r,\delta)$ locality were introduced by
Prakash \emph{et al.} into distributed storage systems (DSSs) due to their
benefit of locally repairing at least $\delta-1$ erasures via other $r$
survival nodes among the same local group. An LRC achieving the $(r,\delta)$
Singleton-type bound is called an optimal $(r,\delta)$ LRC. Constructions of
optimal $(r,\delta)$ LRCs with longer code length and determining the maximal
code length have been an important research direction in coding theory in
recent years. In this paper, we conduct further research on the improvement of
maximum code length of optimal $(r,\delta)$ LRCs. For $2\delta+1\leq d\leq
2\delta+2$, our upper bounds largely improve the ones by Cai \emph{et al.},
which are tight in some special cases. Moreover, we generalize the results of
Chen \emph{et al.} and obtain a complete characterization of optimal $(r=2,
\delta)$-LRCs in the sense of geometrical existence in the finite projective
plane $PG(2,q)$. Within this geometrical characterization, we construct a class
of optimal $(r,\delta)$ LRCs based on the sunflower structure. Both the
construction and upper bounds are better than previous ones.
