  Local Reconstruction Codes (LRCs) allow for recovery from a small number of
erasures in a local manner based on just a few other codeword symbols. A
maximally recoverable (MR) LRC offers the best possible blend of such local and
global fault tolerance, guaranteeing recovery from all erasure patterns which
are information-theoretically correctable given the presence of local recovery
groups. In an $(n,r,h,a)$-LRC, the $n$ codeword symbols are partitioned into
$r$ disjoint groups each of which include $a$ local parity checks capable of
locally correcting $a$ erasures.
  MR LRCs have received much attention recently, with many explicit
constructions covering different regimes of parameters. Unfortunately, all
known constructions require a large field size that exponential in $h$ or $a$,
and it is of interest to obtain MR LRCs of minimal possible field size. In this
work, we develop an approach based on function fields to construct MR LRCs. Our
method recovers, and in most parameter regimes improves, the field size of
previous approaches. For instance, for the case of small $r \ll \epsilon \log
n$ and large $h \ge \Omega(n^{1-\epsilon})$, we improve the field size from
roughly $n^h$ to $n^{\epsilon h}$. For the case of $a=1$ (one local parity
check), we improve the field size quadratically from $r^{h(h+1)}$ to $r^{h
\lfloor (h+1)/2 \rfloor}$ for some range of $r$. The improvements are modest,
but more importantly are obtained in a unified manner via a promising new idea.
