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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

A locally repairable code (LRC) is a <inline-formula> <tex-math notation="LaTeX">$[n, k, d]$ </tex-math></inline-formula> linear code with length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, dimension <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>, minimum distance <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> and locality <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>, which means that every code symbol can be repaired by at most <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> other symbols. LRCs have become an important candidate in distributed storage systems due to their relatively low I/O cost. An LRC is said to be optimal if its minimum distance meets one of the Singleton-like bounds. This paper considers the optimal constructions of LRCs with locality <inline-formula> <tex-math notation="LaTeX">$r=1$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$r=k-1$ </tex-math></inline-formula>, which involves three types: <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-local LRCs, <inline-formula> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula>-LRCs and LRCs with <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-availability. Specifically, we first prove that the existence of an optimal LRC with locality <inline-formula> <tex-math notation="LaTeX">$r=1$ </tex-math></inline-formula> is equivalent to that of an MDS code with certain parameters. Thus we can completely characterize the three types of optimal LRCs with <inline-formula> <tex-math notation="LaTeX">$r=1$ </tex-math></inline-formula> based on some known constructions of MDS codes. Near MDS codes is a special class of sub-optimal linear code whose minimum distance <inline-formula> <tex-math notation="LaTeX">$d=n-k$ </tex-math></inline-formula> and the <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>-th generalized Hamming weight achieves the generalized Singleton bound for <inline-formula> <tex-math notation="LaTeX">$2\leq i \leq k$ </tex-math></inline-formula>. For <inline-formula> <tex-math notation="LaTeX">$r=k-1$ </tex-math></inline-formula>, we have established the connections between optimal <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-local LRCs/ LRCs with <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-availability and near MDS codes. Such connections can help to construct optimal LRCs with <inline-formula> <tex-math notation="LaTeX">$r=k-1$ </tex-math></inline-formula> from some known classes of near MDS codes.

Complete Characterizations of Optimal Locally Repairable Codes With Locality 1 and $K-1$