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

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.