An Improved Sub-Packetization Bound for Minimum Storage Regenerating Codes

Abstract

Distributed storage systems employ codes to provide resilience to failure of multiple storage disks. Specifically, an $(n, k)$ MDS code stores $k$ symbols in $n$ disks such that the overall system is tolerant to a failure of up to $n-k$ disks. However, access to at least $k$ disks is still required to repair a single erasure. To reduce repair bandwidth, array codes are used where the stored symbols or packets are vectors of length $\ell$. MDS array codes have the potential to repair a single erasure using a fraction $1/(n-k)$ of data stored in the remaining disks. We introduce new methods of analysis which capitalize on the translation of the storage system problem into a geometric problem on a set of operators and subspaces. In particular, we ask the following question: for a given $(n, k)$, what is the minimum vector-length or sub-packetization factor $\ell$ required to achieve this optimal fraction? For \emph{exact recovery} of systematic disks in an MDS code of low redundancy, i.e. $k/n > 1/2$, the best known explicit codes \cite{WTB12} have a sub-packetization factor $\ell$ which is exponential in $k$. It has been conjectured \cite{TWB12} that for a fixed number of parity nodes, it is in fact necessary for $\ell$ to be exponential in $k$. In this paper, we provide a new log-squared converse bound on $k$ for a given $\ell$, and prove that $k \le 2\log_2\ell\left(\log_{\delta}\ell+1\right)$, for an arbitrary number of parity nodes $r = n-k$, where $\delta = r/(r-1)$.