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.
翻译：我们在分包装水平($, k, d=n, =n, =n, =1,\alpha) 上提出了两个较低界限,其参数为$(n, k=n, d=n, =1,\alpha),其值为美元,其值为块长度、 美元维度、 在单节修理期间接触的帮助节点数量和分包装水平。 我们提出的第一个约束是用于任何单包装标准, 其值为$(alpha)\ge e e-frac{(k-1)(r-1) { ⁇ 2r ⁇ 2} 。 我们提出的第二个约束是用于最佳获取的MSR代码, 其值为最佳使用量, 并且由 $\ alpha\ ge\ ge\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {r\\ \\ \\ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
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