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In this paper, we revisit the problem of finding the longestsystematic-length $k$ for a linear minimum storage regenerating (MSR) code withoptimal 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 byfollowing a geometric analysis of linear subspaces and operators. First, asimple quadratic bound is given, which implies that $k=r+2$ is the largestnumber of systematic nodes in the \emph{scalar} scenario. Second, an$r$-based-log bound is derived, which is superior to the upper bound onlog-base $2$ in the prior work. Finally, an explicit upper bound depending onthe value of $\frac{r^2}{l}$ is introduced, which further extends thecorresponding result in the literature.

Improved Upper Bounds on Systematic-Length for Linear Minimum Storage Regenerating Codes