On Sequential Locally Repairable Codes

TL;DR

This paper studies sequential locally repairable codes (SLRC) for recovering multiple erasures, providing tight bounds on code rate for t=3 and new constructions that outperform existing codes for t≥4.

Contribution

It derives a tight upper bound on the code rate for t=3 and proposes two novel binary code constructions for arbitrary t≥2, improving upon existing codes.

Findings

Tight upper bound on code rate for t=3.

Two new code constructions for t≥2.

Codes with higher rates than existing families for t≥4.

Abstract

We consider the locally repairable codes (LRC), aiming at sequential recovering multiple erasures. We define the (n,k,r,t)-SLRC (Sequential Locally Repairable Codes) as an [n,k] linear code where any t'(>= t) erasures can be sequentially recovered, each one by r (2<=r<k) other code symbols. Sequential recovering means that the erased symbols are recovered one by one, and an already recovered symbol can be used to recover the remaining erased symbols. This important recovering method, in contrast with the vastly studied parallel recovering, is currently far from understanding, say, lacking codes constructed for arbitrary t>=3 erasures and bounds to evaluate the performance of such codes. We first derive a tight upper bound on the code rate of (n, k, r, t)-SLRC for t=3 and r>=2. We then propose two constructions of binary (n, k, r, t)-SLRCs for general r,t>=2 (Existing constructions are dealing with t<=7 erasures. The first construction generalizes the method of direct product construction. The second construction is based on the resolvable configurations and yields SLRCs for any r>=2 and odd t>=3. For both constructions, the rates are optimal for t in {2,3} and are higher than most of the existing LRC families for arbitrary t>=4.