Multi-Erasure Locally Recoverable Codes Over Small Fields

TL;DR

This paper introduces a new construction of Multi-Erasure Locally Recoverable Codes over small fields using generalized tensor product codes, providing optimal codes with efficient decoding for storage systems.

Contribution

It proposes a general construction method for ME-LRCs over small fields, including bounds, decoding algorithms, and explicit code examples, and relates GII codes to tensor product codes.

Findings

Constructed optimal ME-LRCs over small fields.

Developed a decoding algorithm for erasure recovery.

Established the relation between GII codes and tensor product codes.

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 storage arrays. 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. Our main contribution is to propose a general construction of ME-LRCs based on generalized tensor product codes, and study their erasure-correcting properties. A decoding algorithm tailored for erasure recovery is given, and correctable erasure patterns are identified. We then prove that our construction yields optimal ME-LRCs with a wide range of code parameters, and present some explicit ME-LRCs over small fields. Finally, we show that generalized integrated interleaving (GII) codes can be treated as a subclass of generalized tensor product codes, thus defining the exact relation between these codes.