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Are there any known examples of $ ((n,K,d)) $ codes with $ d \geq 2 $ for which it is not possible to find a basis of codewords that are stabilizer states?

A code word stabilized (CWS) code is defined in https://arxiv.org/abs/0708.1021 and is of this type where, in particular, the different stabilizer state codewords are related by Paulis.

Even though the paper is from 2007 this seems to be the state of the art for non stabilizer codes. At least I can't find any newer papers on the subject.

The paper claims that all known examples of "codes with good parameters" are CWS codes. What is an example of any code (I don't care what the parameters are as long as $ d \geq 2 $) with codewords that are not stabilizer states?

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  • $\begingroup$ One strategy: find an e.g. plane in $\mathbb{C}^2 \otimes \mathbb{C}^2$ that doesn't contain any stabilizer states, and then lift that to the logical space of an $[[n, 2, d]]$ stabilizer code with $d>1$. $\endgroup$
    – squiggles
    Mar 22 at 3:36

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The codes in Permutationally Invariant Codes for Quantum Error Correction as well as the multiqubit codes in section seven of Multispin Clifford codes for angular momentum errors in spin systems and the codes in A Family of Quantum Codes with Exotic Transversal Gates are all $ d \geq 3 $ and permutationally invariant. Thus they are certainly not stabilizer codes by Theorem 3.1 of Investigations on Automorphism Groups of Quantum Stabilizer Codes .

Even stronger, for all these code there are no stabilizer states in the codespace and thus all the codes given above are not CWS. This follows from that fact that all codewords are totally symmetric yet there are no codewords corresponding to a single Dicke state; the codewords are always superpositions of multiple Dicke states.

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