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Quantum Hamming code

A fixed CSS block whose qubits are labeled by the fifteen nonzero vectors of F_2^4. The four coordinate bits form a classical Hamming parity-check matrix, and using that same matrix for both X and Z checks gives eight commuting stabilizers. The code encodes seven qubits and corrects any one physical-qubit Pauli error. It is not the [[15,1,3]] quantum Reed–Muller code: that block is obtained only after six additional independent Z constraints fix six of these logical degrees of freedom.