An Improved Sub-Packetization Bound for Minimum Storage Regenerating Codes

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 \(\delta = r/(r-1)\).