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

An $(n,k,\ell)$-vector MDS code is a $\mathbb{F}$-linear subspace of$(\mathbb{F}^\ell)^n$ (for some field $\mathbb{F}$) of dimension $k\ell$, suchthat any $k$ (vector) symbols of the codeword suffice to determine theremaining $r=n-k$ (vector) symbols. The length $\ell$ of each codeword symbolis called the sub-packetization of the code. Such a code is called minimumstorage regenerating (MSR), if any single symbol of a codeword can be recoveredby downloading $\ell/r$ field elements (which is known to be the leastpossible) from each of the other symbols.  MSR codes are attractive for use in distributed storage systems, and by now avariety of ingenious constructions of MSR codes are available. However, theyall suffer from exponentially large sub-packetization $\ell \gtrsim r^{k/r}$.Our main result is an almost tight lower bound showing that for an MSR code,one must have $\ell \ge \exp(\Omega(k/r))$. This settles a central openquestion concerning MSR codes that has received much attention. Previously, alower bound of $\approx \exp(\sqrt{k/r})$, and a tight lower bound for arestricted class of "optimal access" MSR codes, were known.

An Exponential Lower Bound on the Sub-Packetization of MSR Codes