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

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 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 r(2)/l is introduced, which further extends the corresponding result in the literature.