Example $ GF(4) $ linear $ [[9,1,3]] $ code

Question

Many well known stabilizer codes are $ GF(4) $ linear. For example, the perfect $ [[5,1,3]] $ code and the $ [[7,1,3]] $ Steane code are both $ GF(4) $ linear.

The $ [[9,1,3]] $ Shor code is not $ GF(4) $ linear since it is a CSS code with the number of $ X $ type stabilizer generators different from the number of $ Z $ type stabilizer generators.

Is there some way to modify the Shor code to get a $ GF(4) $ linear $ [[9,1,3]] $ code?

More generally, what is an example of a $ GF(4) $ linear $ [[9,1,3]] $ code?

Answer 1

Here's a possible code. I only checked that it's GF(4) linear and is self orthogonal ; the distance should be 3 by construction. More details in related question Generators for $[[9,1,3]]$ linear quantum code

   [[1,3,2,1,1,1,0,0,0],
     [0,2,1,2,2,0,2,1,0],
     [2,0,1,2,0,3,0,0,0],
     [3,0,3,0,2,3,3,0,1],
     [2,1,3,2,2,2,0,0,0],
     [0,3,2,3,3,0,3,2,0],
     [3,0,2,3,0,1,0,0,0],
     [1,0,1,0,3,1,1,0,2]]

$[0,1,2,3] \leftrightarrow [0,1,\omega,\omega^2]$