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In this paper, we revisit the problem of finding the longest systematic-length $k$ for a linear minimum storage regenerating (MSR) code with optimal repair of only systematic part, for a given per-node storage capacity $l$ and an arbitrary number of parity nodes $r$. We study the problem by following a geometric analysis of linear subspaces and operators. First, a simple quadratic bound is given, which implies that $k=r+2$ is the largest number of systematic nodes in the emph{scalar} scenario. Second, an $r$-based-log bound is derived, which is superior to the upper bound on log-base $2$ in the prior work. Finally, an explicit upper bound depending on the value of $frac{r^2}{l}$ is introduced, which further extends the corresponding result in the literature.
This short note revisits the problem of designing secure minimum storage regenerating (MSR) codes for distributed storage systems. A secure MSR code ensures that a distributed storage system does not reveal the stored information to a passive eavesdr
The $l$-th stopping redundancy $rho_l(mathcal C)$ of the binary $[n, k, d]$ code $mathcal C$, $1 le l le d$, is defined as the minimum number of rows in the parity-check matrix of $mathcal C$, such that the smallest stopping set is of size at least $
A code construction and repair scheme for optimal functional regeneration of multiple node failures is presented, which is based on stitching together short MDS codes on carefully chosen sets of points lying on a linearized polynomial. The nodes are
This chapter deals with the topic of designing reliable and efficient codes for the storage and retrieval of large quantities of data over storage devices that are prone to failure. For long, the traditional objective has been one of ensuring reliabi
In this paper we develop a technique to extend any bound for cyclic codes constructed from its defining sets (ds-bounds) to abelian (or multivariate) codes. We use this technique to improve the searching of new bounds for abelian codes.