The following are my works on distributed storage systems. They study regenerating codes.

- [MoulinAlg20]
I. Duursma, H.-P. Wang.
*Multilinear Algebra for Distributed Storage*. arXiv. - [Atrahasis20]
I. Duursma, X. Li, H.-P. Wang.
*Multilinear Algebra for Minimum Storage Regenerating Codes*. arXiv. - [MoulinAlg21]
I. Duursma, H.-P. Wang.
*Multilinear Algebra for Distributed Storage*. SIAM Journal on Applied Algebra and Geometry (SIAGA). (Journal version of [MoulinAlg20]) - [Atrahasis21]
I. Duursma, X. Li, H.-P. Wang.
*Multilinear Algebra for Minimum Storage Regenerating Codes: A Generalization of Product-Matrix Construction*. Applicable Algebra in Engineering, Communication and Computing. (Journal version of [Atrahasis20])

A **regenerating code** consists of

- a file of size $M$ symbols and
- a system of $n$ storage devices, called
**nodes**.

The configuration of the nodes satisfies the following conditions:

- Each node stores $\alpha$ symbols of the file.
- Any $k$ nodes contains sufficient information to recover the file.
- When a node fails, some $d$ other nodes will each send it $\beta$ symbols to repair the failing node.

The code is named regenerating mainly due to the last bullet pointâ€”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 $M$. [MoulinAlg20] 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 MoulinAlg20 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 MoulinAlg20 that I am still working on.) For more general parameters, check out this D3.js plot.

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.