Multilinear Algebra for Distributed Storage

TL;DR

This paper develops a multilinear algebra framework to construct exact-repair regenerating codes for distributed storage, unifying and extending existing trade-offs and repair strategies.

Contribution

It introduces a multilinear algebra foundation for assembling ERRCs across all relevant parameters, linking trade-offs with cascade codes and addressing multiple failure repairs.

Findings

Established a multilinear algebra foundation for ERRCs.

Connected ERRCs with cascade code trade-offs.

Provided methods for repairing multiple failures.

Abstract

An $(n, k, d, \alpha, \beta, M)$-ERRC (exact-repair regenerating code) is a collection of $n$ nodes used to store a file. For a file of total size $M$, each node stores $\alpha$ symbols, any $k$ nodes recover the file, and any $d$ nodes repair any other node via sending out $\beta$ symbols. We establish a multilinear algebra foundation to assemble $(n, k, d, \alpha, \beta, M)$-ERRCs for all meaningful $(n, k, d)$ tuples. Our ERRCs tie the $\alpha/M$-versus-$\beta/M$ trade-off with cascade codes, the best known construction for this trade-off. We give directions on how these ERRCs repair multiple failures.