  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.
