  Erasure codes play an important role in storage systems to prevent data loss.
In this work, we study a class of erasure codes called Multi-Erasure Locally
Recoverable Codes (ME-LRCs) for storage arrays. Compared to previous related
works, we focus on the construction of ME-LRCs over small fields. We first
develop upper and lower bounds on the minimum distance of ME-LRCs. Our main
contribution is to propose a general construction of ME-LRCs based on
generalized tensor product codes, and study their erasure-correcting
properties. A decoding algorithm tailored for erasure recovery is given, and
correctable erasure patterns are identified. We then prove that our
construction yields optimal ME-LRCs with a wide range of code parameters, and
present some explicit ME-LRCs over small fields. Finally, we show that
generalized integrated interleaving (GII) codes can be treated as a subclass of
generalized tensor product codes, thus defining the exact relation between
these codes.
