Maximally recoverable local reconstruction codes from subspace direct sum systems

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.