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, $d$ is the number of helper nodes contacted during single-node repair, $\alpha $ the sub-packetization level and $k\alpha $ the scalar dimension. 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 $\alpha \ge \min \left\{{ r^{\frac {n-1}{r}}, r^{k-1} }\right\}$ . 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 code to have optimal sub-packetization level under the constraint that the $\beta $ scalar symbol indices we access from a given helper node is dependent only on the index of the failed node, it is necessary that the support of the parity-check matrix be the same as the support structure of the existing MSR constructions in literature such as the Clay code.

中文翻译：

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MSR 码子分组级别的下界和表征最优接入 MSR 码的特性

我们提出了子分组级别的两个下界 $\阿尔法$带参数的 MSR 代码 $(n, k, d=n-1, \alpha )$在哪里 $n$是块长度， $d$是单节点修复期间联系的辅助节点数， $\阿尔法$子分组级别和 $k\阿尔法$标量维度。我们提出的第一个界限适用于任何 MSR 代码，由下式给出 $\alpha \ge e^{\frac {(k-1)(r-1)}{2r^{2}}}$ . 我们提出的第二个界限是针对最佳访问 MSR 码的情况，界限由下式给出 $\alpha \ge \min \left\{{ r^{\frac {n-1}{r}}, r^{k-1} }\right\}$ . 存在最佳访问 MSR 结构，可实现第二个子分组级别的限制，该等式限制使该限制紧密。我们还证明，对于最优访问 MSR 码，在以下约束条件下具有最优子分组级别： $\beta $我们从给定辅助节点访问的标量符号索引仅取决于故障节点的索引，奇偶校验矩阵的支持必须与文献中现有 MSR 结构的支持结构相同，例如粘土代码。