Quantum dual extended Hamming code immune to collective coherent errors

Collective coherent (CC) errors are inevitable, as every physical qubit undergoes free evolution under its kinetic Hamiltonian. These errors can be more damaging than stochastic Pauli errors because they affect all qubits coherently, resulting in high-weight errors that standard quantum error-correcting (QEC) codes struggle to correct. In quantum memories and communication systems, especially when storage durations are long, CC errors often dominate over stochastic noise. Trapped-ion platforms, for example, exhibit strong CC errors with minimal stochastic Pauli components. In this work, we address the regime where immunity to CC errors, high code rate (due to limited qubit availability), and moderate distance (sufficient for correcting low-weight errors) are all essential. We construct a family of constant-excitation stabilizer codes with parameters $[[{2}^{r+1},{2}^{r}\ensuremath{-}(r+1),4]]$. The smallest instance, the [[8,1,4]] code, improves the code rate and error threshold of the best previously known CE code by factors of approximately two and four, respectively.