High-rate amplitude-damping Shor codes with immunity to collective coherent errors

We introduce a family of high-rate amplitude-damping (AD) Shor codes, designed to effectively correct AD errors while maintaining immunity to collective coherent (CC) errors. The proposed $[[(w+1)(w+K),K]]$ AD codes can approximately correct up to $w$ AD errors, with flexible parameters $(w,K)$, and we provide a rigorous proof that these codes satisfy the approximate quantum error-correction conditions. These codes leverage structured stabilizer measurements, enabling efficient detection of AD errors using only local operations and ancillary qubits. We further construct a family of CC-AD Shor codes by concatenating these AD codes with the dual-rail code.