The following are my works on distributed storage systems.
|MoulinAlge20||Multilinear Algebra for Distributed Storage|
|Atrahasis20||Multilinear Algebra for Minimum Storage Regenerating Codes|
A regenerating code consists of
The configuration of the nodes satisfies the following conditions:
The code is named regenerating mainly due to the last bullet point—the nodes regenerate themselves.
The theory of regenerating codes concerns the relation among $n, k, d, \alpha, \beta, M$. For example, since any $k$ nodes contain $k\alpha$ symbols and can recover the file, the file size $M$ is at most $k\alpha$. Similarly, since $d\beta$ symbols repair a failing node, the node size $\alpha$ is at most $d\beta$. (Exercise) One can also show that $k - 1$ nodes ($\alpha$) plus $d - k + 1$ help messages ($\beta$) is at least $M$. There is a family of bounds of this type. They are called cut-set bounds and restrict where those parameters can live.
The opposite approach is to construct regenerating codes that aim to achieve low $\alpha$, low $\beta$, and high $M$. MoulinAlge20 utilizes multilinear algebra to do this. We construct a series of regenerating codes which we call moulin codes. They achieve the best known $\alpha/M$-versus-$\beta/M$ trade-off to date. And it is conjectured that this trade-off is optimal.
See Figure 1 on page 3 in MoulinAlge20 for the $\alpha/M$-versus-$\beta/M$ trade-off for the $(n, 3, 3)$ case. Here is another $\alpha/M$-versus-$\beta/M$ trade-off for the $(n, 3, 4)$ case. (In a newer version of MoulinAlge20 that I am still working on.)
See also Table 2 on page 29 for the relations among some competitive constructions.
Atrahasis20 exploits multilinear algebra to construct MSR codes, which we called Atrahasis codes. Formally, an MSR code is a regenerating code with $M = k\alpha$ and $\beta = \alpha/(d - k + 1)$. From the constraint on $M$ one sees that there is no wastes of storage (hence the name minimum storage regeneration = MSR). Some researchers see MSR codes as the intersection of regenerating codes and MDS codes.
MSR alone attracts significant attentions because people want to minimize node size ($\alpha \geq M/k$), and only then they minimize help messages ($\beta \geq \alpha/(d - k + 1)$ given that $\alpha \geq M/k$). See Table 1 on page 5 in Atrahasis20 for a comparison of some existing contraptions.