Quantum $(r,\delta)$-Locally Recoverable BCH and Homothetic-BCH Codes

TL;DR

This paper develops quantum locally recoverable codes based on BCH and homothetic-BCH codes, achieving optimality for large-scale storage systems by ensuring efficient recovery of multiple failures.

Contribution

It introduces a method to construct quantum $(r, ext{delta})$-LRCs from BCH and homothetic-BCH codes, resulting in optimal codes meeting the Singleton-like bound.

Findings

Constructed quantum $(r, ext{delta})$-LRCs from classical BCH codes.

Achieved optimal quantum LRCs that meet the Singleton-like bound.

Provided a framework for quantum codes suitable for distributed storage.

Abstract

Quantum $(r,\delta)$-locally recoverable codes ($(r,\delta)$-LRCs) are the quantum version of classical $(r,\delta)$-LRCs designed to recover multiple failures in large-scale distributed and cloud storage systems. A quantum $(r,\delta)$-LRC, $Q(C)$, can be constructed from an $(r,\delta)$-LRC, $C$, which is Euclidean or Hermitian dual-containing. This article is devoted to studying how to get quantum $(r,\delta)$-LRCs from BCH and homothetic-BCH codes. As a consequence, we give pure quantum $(r,\delta)$-LRCs which are optimal for the Singleton-like bound.