Repair Locality with Multiple Erasure Tolerance

TL;DR

This paper introduces a new locality concept for erasure codes in distributed storage, enhancing repair efficiency and erasure tolerance, and provides bounds and constructions demonstrating improved minimum distance and high information rate.

Contribution

It proposes the $(r, ext{ extdelta})_c$-locality, derives bounds on minimum distance, and constructs codes with improved distance and high rate compared to existing locality definitions.

Findings

Codes with $(r, extdelta)_c$-locality have higher minimum distance.

Existence of codes attains the derived bounds for large $n$.

Gain in minimum distance is proportional to $ extsqrt{r}$.

Abstract

In distributed storage systems, erasure codes with locality $r$ is preferred because a coordinate can be recovered by accessing at most $r$ other coordinates which in turn greatly reduces the disk I/O complexity for small $r$. However, the local repair may be ineffective when some of the $r$ coordinates accessed for recovery are also erased. To overcome this problem, we propose the $(r,\delta)_c$-locality providing $\delta -1$ local repair options for a coordinate. Consequently, the repair locality $r$ can tolerate $\delta-1$ erasures in total. We derive an upper bound on the minimum distance $d$ for any linear $[n,k]$ code with information $(r,\delta)_c$-locality. For general parameters, we prove existence of the codes that attain this bound when $n\geq k(r(\delta-1)+1)$, implying tightness of this bound. Although the locality $(r,\delta)$ defined by Prakash et al provides the same level of locality and local repair tolerance as our definition, codes with $(r,\delta)_c$-locality are proved to have more advantage in the minimum distance. In particular, we construct a class of codes with all symbol $(r,\delta)_c$-locality where the gain in minimum distance is $\Omega(\sqrt{r})$ and the information rate is close to 1.