Maximally Recoverable LRCs: A Field Size Lower Bound and Constructions for Few Heavy Parities

Abstract

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 F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ω(h)</sup> 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 ≥ h, while r may grow with n, our lower bound simplifies to q ≥ Ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a,h</sub> (n · r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min{a,</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h-2}</sup> ) . 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 ≤ 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 ≥ 3. When h = 3, we give a construction with O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) field size, whereas previous constructions needed n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Θ(a)</sup> 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.