# Uniform Minors in Maximally Recoverable Codes

## TL;DR

This paper investigates the structure of maximally recoverable locally recoverable codes using matroid theory, identifying uniform minors to improve bounds on the necessary field size for such codes.

## Contribution

It introduces a method to classify uniform minors in maximally recoverable codes, leading to tighter bounds on the field size requirements.

## Key findings

- Derived the complete list of uniform minors in these codes.
- Improved the non-asymptotic lower bound on field size.
- Connected matroid theory with code recoverability properties.

## Abstract

In this letter, locally recoverable codes with maximal recoverability are studied with a focus on identifying the MDS codes resulting from puncturing and shortening. By using matroid theory and the relation between MDS codes and uniform minors, the list of all the possible uniform minors is derived. This list is used to improve the known non-asymptotic lower bound on the required field size of a maximally recoverable code.