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.
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