  Maximally Recoverable Local Reconstruction Codes (LRCs) are codes designed
for distributed storage to provide maximum resilience to failures for a given
amount of storage redundancy and locality. An $(n,r,h,a,g)$-MR LRC has $n$
coordinates divided into $g$ local groups of size $r=n/g$, where each local
group has `$a$' local parity checks and there are an additional `$h$' global
parity checks. Such a code can correct `$a$' erasures in each local group and
any $h$ additional erasures. Constructions of MR LRCs over small fields is
desirable since field size determines the encoding and decoding efficiency in
practice. In this work, we give a new construction of $(n,r,h,a,g)$-MR-LRCs
over fields of size $q=O(n)^{h+(g-1)a-\lceil h/g\rceil}$ which generalizes a
construction of Hu and Yekhanin (ISIT 2016). This improves upon state of the
art when there are a small number of local groups, which is true in practical
deployments of MR LRCs.
