  Maximally recoverable local reconstruction codes (MR LRCs for short) have
received great attention in the last few years. Various constructions have been
proposed in literatures. The main focus of this topic is to construct MR LRCs
over small fields. An $(N=nr,r,h,\Gd)$-MR LRC is a linear code over finite
field $\F_\ell$ of length $N$, whose codeword symbols are partitioned into $n$
local groups each of size $r$. Each local group can repair any $\Gd$ erasure
errors and there are further $h$ global parity checks to provide fault
tolerance from more global erasure patterns.
  MR LRCs deployed in practice have a small number of global parities such as
$h=O(1)$. In this parameter setting, all previous constructions require the
field size $\ell =\Omega_h (N^{h-1-o(1)})$. It remains challenging to improve
this bound. In this paper, via subspace direct sum systems, we present a
construction of MR LRC with the field size $\ell= O(N^{h-2+\frac1{h-1}-o(1)})$.
In particular, for the most interesting cases where $h=2,3$, we improve
previous constructions by either reducing field size or removing constraints.
In addition, we also offer some constructions of MR LRCs for larger global
parity $h$ that have field size incomparable with known upper bounds. The main
techniques used in this paper is through subspace direct sum systems that we
introduce. Interestingly, subspace direct sum systems are actually equivalent
to $\F_q$-linear codes over extension fields. Based on various constructions of
subspace direct sum systems, we are able to construct several classes of MR
LRCs.
