Multilinear Algebra for Minimum Storage Regenerating Codes

Abstract

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 recover the file, and any $d$ nodes can repair any other node via each sending out $\alpha/(d-k+1)$ symbols. In this work, we explore various ways to re-express the infamous product-matrix construction using skew-symmetric matrices, polynomials, symmetric algebras, and exterior algebras. We then introduce a multilinear algebra foundation to produce $\bigl(n, k, \frac{(k-1)t}{t-1}, \binom{k-1}{t-1}\bigr)$-MSR codes for general $t\geq2$. At the $t=2$ end, they include the product-matrix construction as a special case. At the $t=k$ end, we recover determinant codes of mode $m=k$; further restriction to $n=k+1$ makes it identical to the layered code at the MSR point. Our codes' sub-packetization level---$\alpha$---is independent of $n$ and small. 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 subset of practical parameters. We offer hints on how our code repairs multiple failures at once.