On Lower Bounds on Sub-Packetization Level of MSR codes and On The Structure of Optimal-Access MSR Codes Achieving The Bound

We present two lower bounds on sub-packetization level $\alpha$ of MSR codes with parameters $(n, k, d=n-1, \alpha)$ where $n$ is the block length, $k$ dimension, $d$ number of helper nodes contacted during single node repair and $\alpha$ the sub-packetization level. The first bound we present is for any MSR code and is given by $\alpha \ge e^{\frac{(k-1)(r-1)}{2r^2}}$. The second bound we present is for the case of optimal-access MSR codes and the bound is given by $\alpha \ge \min \{ r^{\frac{n-1}{r}}, r^{k-1} \}$. There exist optimal-access MSR constructions that achieve the second sub-packetization level bound with an equality making this bound tight. We also prove that for an optimal-access MSR codes to have optimal sub-packetization level under the constraint that the indices of helper symbols are dependant only on the failed node, it is needed that the support of the parity check matrix is same as the support structure of several other optimal constructions in literature.