# Hsin-Po Wang

Math PhD @ UIUC

From the oldest to the newest.

Abbreviation Title
MoulinAlge20 Multilinear Algebra for Distributed Storage
Atrahasis20 Multilinear Algebra for Minimum Storage Regenerating Codes

Both MoulinAlge20 and Atrahasis20 concern regenerating codes that have applications in distributed storage systems.

A regenerating code consists of

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

They satisfy 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, any $d$ other nodes will each sends it $\beta$ symbols to repair.

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 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 restrict where those parameters can live.

The opposite approach is to construct regenerating codes that aim to achieve low $\alpha$ and $\beta$ and high $M$. MoulinAlge20 uses 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. It is conjectured that this trade-off is optimal.

See Figure 1 on page 3 in MoulinAlge20 for an example of $\alpha/M$-versus-$\beta/M$ trade-off.

See also Table 2 on page 29 for the relations among some competitive constructions.

Atrahasis20 uses multilinear algebra to construct MSR 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 constructions.