  Aiming to recover the data from several concurrent node failures, linear
$r$-LRC codes with locality $r$ were extended into $(r, \delta)$-LRC codes with
locality $(r, \delta)$ which can enable the local recovery of a failed node in
case of more than one node failure. Optimal LRC codes are those whose
parameters achieve the generalized Singleton bound with equality. In the
present paper, we are interested in studying optimal LRC codes over small
fields and, more precisely, over $\mathbb{F}_4$. We shall adopt an approach by
investigating optimal quaternary $(r,\delta)$-LRC codes through their
parity-check matrices.
  Our study includes determining the structural properties of optimal
$(r,\delta)$-LRC codes, their constructions, and their complete classification
over $\F_4$ by browsing all possible parameters. We emphasize that the precise
structure of optimal quaternary $(r,\delta)$-LRC codes and their classification
are obtained via the parity-check matrix approach use proofs-techniques
different from those used recently for optimal binary and ternary
$(r,\delta)$-LRC codes obtained by Hao et al. in [IEEE Trans. Inf. Theory,
2020, 66(12): 7465-7474].
