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 systems 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 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 $r$. It has $h$ global parity checks and each local group has $a$ local parity checks. 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. We show the first non-trivial lower bounds on the field size required for MR LRCs. When $a,h$ are constant and the number of local groups $g \ge h$, while $r$ may grow with $n$, our lower bound simplifies to $q\ge Ω_{a,h}\left(n\cdot r^{\min\{a,h-2\}}\right).$ No superlinear (in $n$) lower bounds were known prior to this work for any setting of parameters. 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 $h\le 3$. For $h=2$, we give a linear field size construction. We also show a surprising application of elliptic curves and arithmetic progression free sets in the construction of MR LRCs.