Lower Bounds for Maximally Recoverable Tensor Code and Higher Order MDS Codes

TL;DR

This paper establishes lower bounds on field sizes for maximally recoverable tensor codes with a focus on the case where columns have a single parity check, introducing higher order MDS codes and characterizing correctability.

Contribution

It introduces higher order MDS codes, characterizes MR tensor codes with $a=1$, and derives field size lower bounds, along with a correctability checking algorithm.

Findings

MR tensor codes with $a=1$ require large fields, $q= ext{Omega}(n^{b-1})$

Higher order MDS codes generalize classical MDS codes and relate to MR tensor codes

A polynomial-time algorithm checks correctability of erasure patterns in $a=1$ case

Abstract

An $(m,n,a,b)$-tensor code consists of $m\times n$ matrices whose columns satisfy `$a$' parity checks and rows satisfy `$b$' parity checks (i.e., a tensor code is the tensor product of a column code and row code). Tensor codes are useful in distributed storage because a single erasure can be corrected quickly either by reading its row or column. Maximally Recoverable (MR) Tensor Codes, introduced by Gopalan et al., are tensor codes which can correct every erasure pattern that is information theoretically possible to correct. The main questions about MR Tensor Codes are characterizing which erasure patterns are correctable and obtaining explicit constructions over small fields. In this paper, we study the important special case when $a=1$, i.e., the columns satisfy a single parity check equation. We introduce the notion of higher order MDS codes (MDS$(\ell)$ codes) which is an interesting generalization of the well-known MDS codes, where $\ell$ captures the order of genericity of points in a low-dimensional space. We then prove that a tensor code with $a=1$ is MR iff the row code is an MDS$(m)$ code. We then show that MDS$(m)$ codes satisfy some weak duality. Using this characterization and duality, we prove that $(m,n,a=1,b)$-MR tensor codes require fields of size $q=\Omega_{m,b}(n^{\min\{b,m\}-1})$. Our lower bound also extends to the setting of $a>1$. We also give a deterministic polynomial time algorithm to check if a given erasure pattern is correctable by the MR tensor code (when $a=1$).