# Multi-Erasure Locally Recoverable Codes Over Small Fields For Flash Memory Array

## TL;DR

This paper introduces a new construction of Multi-Erasure Locally Recoverable Codes (ME-LRCs) over small fields for flash memory, providing bounds, decoding algorithms, and optimal code families to enhance data reliability.

## Contribution

It proposes a general construction of ME-LRCs based on generalized tensor product codes, focusing on small field sizes and optimal erasure correction capabilities.

## Key findings

- Derived bounds on minimum distance considering field size
- Constructed optimal ME-LRCs with various parameters
- Presented decoding algorithms for erasure recovery

## Abstract

Erasure codes play an important role in storage systems to prevent data loss. In this work, we study a class of erasure codes called Multi-Erasure Locally Recoverable Codes (ME-LRCs) for flash memory array. Compared to previous related works, we focus on the construction of ME-LRCs over small fields. We first develop upper and lower bounds on the minimum distance of ME-LRCs. These bounds explicitly take the field size into account. Our main contribution is to propose a general construction of ME-LRCs based on generalized tensor product codes, and study their erasure-correcting property. A decoding algorithm tailored for erasure recovery is given. We then prove that our construction yields optimal ME-LRCs with a wide range of code parameters. Finally, we present several families of ME-LRCs over different fields.