I was thinking of an error correcting code to correct 1-qubit errors. I came up with the following, which I guess has to have a mistake somewhere, but I am not able to find it.
The code is the same as the 9 qubit Shor code with one small difference. As in the Shor code, first we encode our qubit using the 3-qubit phase code. Then, instead of further encoding the 3 qubits against bit-flip errors, we only encode one of them, namely, the one that contains the state that we want to protect. The resulting code would be the following $$\small{|0\rangle \rightarrow |00000\rangle + |00001\rangle + |00010\rangle + |11100\rangle + |00011\rangle + |11101\rangle + |11110\rangle + |11111\rangle}$$
$$\small{|1\rangle \rightarrow |00000\rangle - |00001\rangle - |00010\rangle - |11100\rangle + |00011\rangle + |11101\rangle + |11110\rangle - |11111\rangle}$$
Thank you!
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's with\rangle
's, as that's the appropriate notation while denoting kets. $\endgroup$