Maximally Recoverable LRCs: A field size lower bound and constructions for few heavy parities

The explosion in the volumes of data being stored online has resulted in distributed storage ‘s transitioning to erasure coding based schemes. Local Reconstruction Codes (LRCs) have emerged as the codes of choice for these applications. These codes can correct a small number of erasures (which is the typical case) by accessing only a small number of remaining coordinates. An $(n,r,h,a,q)$ -LRC is a linear code over $\mathbb {F}_{q}$ of length $n$ , whose codeword symbols are partitioned into $g=n/r$ local groups each of size $r$ . Each local group has $a$ local parity checks that allow recovery of up to $a$ erasures within the group by reading the unerased symbols in the group. There are a further $h$ “heavy” parity checks to provide fault tolerance from more global erasure patterns. Such an LRC is Maximally Recoverable (MR), if it corrects all erasure patterns which are information-theoretically correctable under the stipulated structure of local and global parity checks, namely patterns with up to $a$ erasures in each local group and an additional $h$ (or fewer) erasures anywhere in the codeword. The existing constructions require fields of size $n^{\Omega (h)}$ while no superlinear lower bounds were known for any setting of parameters. Is it possible to get linear field size similar to the related MDS codes (e.g., Reed-Solomon codes)? In this work, we answer this question by showing superlinear lower bounds on the field size of MR-LRCs. When $a,h$ are constant and the number of local groups $g \geqslant h$ , while $r$ may grow with $n$ , our lower bound simplifies to $q \geqslant \Omega _{a,h}\left ({n\cdot r^{\min \{a,h-2\}}}\right)$ . MR-LRCs deployed in practice have a small number of global parities, typically $h=2,3$ . We complement our lower bounds by giving constructions with small field size for $h \leqslant 3$ . When $h=2$ , we give a linear field size construction, whereas previous constructions required quadratic field size in some parameter ranges. Note that our lower bound is superlinear only if $h \geqslant 3$ . When $h=3$ , we give a construction with $O(n^{3})$ field size, whereas previous constructions needed $n^{\Theta (a)}$ field size. This makes the choices $r=3, a=1, h=3$ the next simplest non-trivial setting to investigate regarding the existence of MR-LRCs over fields of near-linear size. We answer this question in the positive via a novel approach based on elliptic curves and arithmetic progression free sets.