An \((n, k, d, \alpha )\)-MSR (minimum storage regeneration) code is a set of n nodes used to store a file. For a file of total size \(k\alpha\), each node stores \(\alpha\) symbols, any k nodes determine the file, and any d nodes can repair any other node by each sending out \(\alpha /(d-k+1)\) symbols. In this work, we express the product-matrix construction of \(\bigl (n, k, 2(k-1), k-1\bigr )\)-MSR codes in terms of symmetric algebras. We then generalize the product-matrix construction to \(\bigl (n, k, \frac{(k-1)t}{t-1}, \left( {\begin{array}{c}k-1\\ t-1\end{array}}\right) \bigr )\)-MSR codes for general \(t\geqslant 2\), while the \(t=2\) case recovers the product-matrix construction. Our codes’ sub-packetization level—\(\alpha\)—is small and independent of n. It is less than \(L^{2.8(d-k+1)}\), where L is Alrabiah–Guruswami’s lower bound on \(\alpha\). Furthermore, it is less than other MSR codes’ \(\alpha\) for a set of practical parameters. Finally, we discuss how our code repairs multiple failures at once.

一个\（（N，K，d，\阿尔法）\） -MSR（最小存储再生）码是一组Ñ用于存储文件的节点。对于总大小为\(k\alpha\) 的文件，每个节点存储\(\alpha\)符号，任意k 个节点确定文件，任意d 个节点可以通过每次发送\(\alpha / (d-k+1)\)符号。在这项工作中，我们根据对称代数来表达\(\bigl (n, k, 2(k-1), k-1\bigr )\) -MSR 代码的乘积矩阵构造。然后我们将乘积矩阵构造推广到\(\bigl (n, k, \frac{(k-1)t}{t-1}, \left( {\begin{array}{c}k-1\ \ t-1\end{array}}\right) \bigr )\)-MSR 代码用于一般\(t\geqslant 2\)，而\(t=2\)情况恢复乘积矩阵构造。我们代码的子包化级别—— \(\alpha\) ——很小并且与n无关。它小于\(L^{2.8(d-k+1)}\)，其中L是 Alrabiah–Guruswami 在\(\alpha\)上的下界 。此外，对于一组实际参数，它小于其他 MSR 代码的\(\alpha\)。最后，我们讨论我们的代码如何一次修复多个故障。