Recognizing a CWS code in a specific case

Question

A CWS code can be defined in terms of stabilizer codes/stabilizer states and graph states. See What is an example of a non-additive code that is not a CWS code?

Is the $ ((11,2,3)) $ nonadditive quantum error-correcting code given in On the Structure of Additive Quantum Codes and the Existence of Nonadditive Codes a codeword stabilized code?

Certainly all the coefficients are $ \pm 1 $ up to a global scalar (in fact all the nonzero coefficients are $ +1 $) so that necessary condition is met.

Maybe there is some way of confirming this (or deriving a contradiction) by thinking about what graph the $ ((11,2,3)) $ code would have to correspond to?

Answer 1

There exists a CWS code with parameters ((11,2,5)) with a graph specified in 1, p. 89.

Answer 2

Theorem 6 of https://arxiv.org/abs/0803.3232 states "All $((n, 2, d))$ CWS codes are additive."

The $ ((11,2,3)) $ code described in the question is non-additive, which can be verified by observing that the weight enumerator coefficients $$ A = (1,0,0,0,\frac{110}{3},0,88,0,605,0,\frac{880}{3},0) \\ B = (1, 0, 0, \frac{55}{3}, \frac{110}{3}, 88, 88, 1210, 605, \frac{4400}{3}, \frac{880}{3}, 289 ) $$ are not all integers. Thus the code is not CWS.