English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In distributed storage systems, locally repairable codes (LRCs) are designedto reduce disk I/O and repair costs by enabling recovery of each code symbolfrom 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 asignificant research topic over the past decade. In \cite{Luo2022}, Luo\emph{et al.} proposed a construction of linear codes by using unions of someprojective subspaces within a projective space. Several new classes of Griesmercodes and distance-optimal codes were constructed, and some of them were provedto 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 intersectingprojective subspaces. This modification enables us to construct good codes withmore flexible parameters. Additionally, we present the conditions for theconstructed linear codes to qualify as Griesmer codes or achieve distanceoptimality. Next, we explore the locality of linear codes constructed byeliminating elements from a complete projective space. The novelty of our worklies 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 whichmay be either Griesmer codes or not Griesmer codes. Finally, we investigate theavailability and alphabet-optimality of $(r,\delta)$-LRCs constructed from ourmodified framework.

Some new constructions of optimal linear codes and alphabet-optimal $(r,\delta)$-locally repairable codes