Improved Constructions and Lower Bounds for Maximally Recoverable Grid Codes
Joshua Brakensiek, Manik Dhar, Sivakanth Gopi
TL;DR
This work studies maximally recoverable grid codes on an $m\times n$ topology with row/column parity checks and $h$ global parities, focusing on the practically relevant regime $a=b=1$ with constant $m,h$ and growing $n$. It develops several explicit constructions that achieve MR with field size poly$(n)$, notably a Gabidulin-based lifting, BCH-based, binary-encoding, and an additive-combinatorics approach for $m=3,h=1$, and it provides new lower bounds showing these poly$(n)$ scalings are close to optimal in this regime. A central tool is the cycle-sum independence criterion, which reduces MR verification to linear (in)dependence of cycle sums associated with cycles formed by adding $h$ chords to a spanning tree. The results have practical implications for storage systems needing robust erasure resilience across large grids while keeping field sizes manageable, and they open questions about the exact dependence of field size on parameters $(m,h)$ in various regimes.
Abstract
In this paper, we continue the study of Maximally Recoverable (MR) Grid Codes initiated by Gopalan et al. [SODA 2017]. More precisely, we study codes over an $m \times n$ grid topology with one parity check per row and column of the grid along with $h \ge 1$ global parity checks. Previous works have largely focused on the setting in which $m = n$, where explicit constructions require field size which is exponential in $n$. Motivated by practical applications, we consider the regime in which $m,h$ are constants and $n$ is growing. In this setting, we provide a number of new explicit constructions whose field size is polynomial in $n$. We further complement these results with new field size lower bounds.