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

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.

24 pages, 6 tables. Version 2 improves the code construction and expands the set of realizable parameters. It also adds comprehensive tables comparing realizable parameters with those known to the authors from previous literature on multiple erasure correction as of July 2026