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A code is a non-stabilizer code if it is not equivalent by local unitaries to a stabilizer code.

What is an example of a non-stabilizer code with distance $ d \geq 2 $?

Are there any non-stabilizer codes that are known to have especially good properties, for example better parameters than any known stabilizer code?

If you have a code in mind that seems like it is non-stabilizer then feel free to share! (even if you don't have a proof that it's not equivalent to a stabilizer code by local unitaries, I'm mostly just interested in seeing code constructions that seem essentially different from stabilizer codes).

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You're looking for non-additive quantum codes. There are many examples but I'll refer you to the seminal paper by Rains, Hardin, Shor & Sloane: https://arxiv.org/abs/quant-ph/9703002.

There's also a more recent framework by Grassl and Roetteler.

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