  As an important coding scheme in modern distributed storage systems, locally
repairable codes (LRCs) have attracted a lot of attentions from perspectives of
both practical applications and theoretical research. As a major topic in the
research of LRCs, bounds and constructions of the corresponding optimal codes
are of particular concerns.
  In this work, codes with $(r,\delta)$-locality which have optimal minimal
distance w.r.t. the bound given by Prakash et al. \cite{Prakash2012Optimal} are
considered. Through parity check matrix approach, constructions of both optimal
$(r,\delta)$-LRCs with all symbol locality ($(r,\delta)_a$-LRCs) and optimal
$(r,\delta)$-LRCs with information locality ($(r,\delta)_i$-LRCs) are provided.
As a generalization of a work of Xing and Yuan \cite{XY19}, these constructions
are built on a connection between sparse hypergraphs and optimal
$(r,\delta)$-LRCs. With the help of constructions of large sparse hypergraphs,
the length of codes constructed can be super-linear in the alphabet size. This
improves upon previous constructions when the minimal distance of the code is
at least $3\delta+1$. As two applications, optimal H-LRCs with super-linear
length and GSD codes with unbounded length are also constructed.
