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].

翻译：为了从几个同时节点失灵中恢复数据,对地方值为$的线性美元-LRC代码,扩大到美元(r,\delta)美元-LRC代码,以(r,\delta)美元-LRC代码,以便在不止一个节点失灵的情况下,能够在当地恢复一个失败节点。最佳LRC代码是那些其参数实现了普遍单一吨的单一值与平等的标准。在本文件中,我们有兴趣研究关于小领域的最佳LRC代码,更确切地说,超过美元(mathbb{F)4美元。我们将采取一种方法,通过对美元(r,\delta)-LRC代码进行对美元(美元)-LRC代码的对等检查。我们的研究包括确定美元-LRC代码、其构造的结构属性,以及通过浏览所有可能的参数,将其完全分类超过$\F_4美元。我们强调,最佳基价(r,r,\delta)$47-LRC的代码及其分类的精确结构,是通过对美元(HA-Bin-chro)采用最佳核对方法,最近使用了标准。