Binary Cyclic Codes that are Locally Repairable

TL;DR

This paper introduces new binary cyclic codes optimized for local repair in storage systems, achieving optimal dimension for given distance and locality, with constructions for multiple repair sets.

Contribution

The paper constructs new binary cyclic codes with optimal parameters for local repair, establishing new bounds and extending to multiple repair sets.

Findings

Constructed cyclic codes with locality 2 and distances 2, 6, 10.

Discovered new upper bounds on code dimension.

Proved optimality of local repair via disjoint groups.

Abstract

Codes for storage systems aim to minimize the repair locality, which is the number of disks (or nodes) that participate in the repair of a single failed disk. Simultaneously, the code must sustain a high rate, operate on a small finite field to be practically significant and be tolerant to a large number of erasures. To this end, we construct new families of binary linear codes that have an optimal dimension (rate) for a given minimum distance and locality. Specifically, we construct cyclic codes that are locally repairable for locality 2 and distances 2, 6 and 10. In doing so, we discover new upper bounds on the code dimension, and prove the optimality of enabling local repair by provisioning disjoint groups of disks. Finally, we extend our construction to build codes that have multiple repair sets for each disk.