In distributed storage systems, locally repairable codes (LRCs) are designed to reduce disk I/O and repair costs by enabling recovery of each code symbol from a small number of other symbols. To handle multiple node failures, $(r,\delta)$-LRCs are introduced to enable local recovery in the event of up to $\delta-1$ failed nodes. Constructing optimal $(r,\delta)$-LRCs has been a significant research topic over the past decade. In \cite{Luo2022}, Luo \emph{et al.} proposed a construction of linear codes by using unions of some projective subspaces within a projective space. Several new classes of Griesmer codes and distance-optimal codes were constructed, and some of them were proved to be alphabet-optimal $2$-LRCs. In this paper, we first modify the method of constructing linear codes in \cite{Luo2022} by considering a more general situation of intersecting projective subspaces. This modification enables us to construct good codes with more flexible parameters. Additionally, we present the conditions for the constructed linear codes to qualify as Griesmer codes or achieve distance optimality. Next, we explore the locality of linear codes constructed by eliminating elements from a complete projective space. The novelty of our work lies in establishing the locality as $(2,p-2)$, $(2,p-1)$, or $(2,p)$-locality, in contrast to the previous literature that only considered $2$-locality. Moreover, by combining analysis of code parameters and the C-M like bound for $(r,\delta)$-LRCs, we construct some alphabet-optimal $(2,\delta)$-LRCs which may be either Griesmer codes or not Griesmer codes. Finally, we investigate the availability and alphabet-optimality of $(r,\delta)$-LRCs constructed from our modified framework.

翻译：在分布式存储系统中，局部修复码（LRC）旨在通过允许每个码符号从少量其他符号恢复，从而降低磁盘I/O和修复成本。为应对多节点故障，引入了$(r,\delta)$-LRC以支持最多$\delta-1$个节点失效时的局部恢复。过去十年间，构造最优$(r,\delta)$-LRC一直是重要的研究课题。在文献\cite{Luo2022}中，Luo等人提出了一种利用射影空间中若干射影子空间并集构造线性码的方法，并构建了若干新类别的Griesmer码与距离最优码，其中部分被证明是字母表最优的2-LRC。本文首先通过考虑射影子空间相交的更一般情形，改进了\cite{Luo2022}中的线性码构造方法。这一改进使我们能够构造参数更灵活的优良码，同时给出了所构造线性码成为Griesmer码或达到距离最优性的条件。其次，我们探讨了通过从完备射影空间中删除元素构造线性码的局部性。本文的创新之处在于建立了$(2,p-2)$、$(2,p-1)$或$(2,p)$-局部性，而此前文献仅考虑2-局部性。此外，结合码参数分析与$(r,\delta)$-LRC的C-M型界，我们构造了若干字母表最优的$(2,\delta)$-LRC，这些码可能属于也可能不属于Griesmer码。最后，我们研究了基于改进框架构造的$(r,\delta)$-LRC的可用性与字母表最优性。