On Sequential Locally Repairable Codes

Abstract

We consider the locally repairable codes (LRCs), aiming at sequentially recovering multiple erasures; in particular, we propose and study the so-called (n, k, r, t)-sequential LRCs (SLRC) as an [n, k] linear code, where any t' (≤ t) erasures can be sequentially recovered, each by r (2 ≤ r <; k) other code symbols. Here, 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 extensively studied parallel recovering, is currently far from being thoroughly understood; more specifically, there are to date no 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 the (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 only deal 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 odd t ≥ 3. For both constructions, the rates are optimal for t ∈ {2, 3} and are higher than most of the existing LRC families for arbitrary t ≥ 4.