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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?

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  • $\begingroup$ codetables.de/QECC.php?q=4&n=9&k=1 gives such an example. However I wasn't able to check that the code is in fact GF(4) linear. The site uses a different convention than (Y=iXZ vs my Y=XZ) so that could be the source of the discrepancy...note that it's an extension of the [[5,1,3]] code : the first 3 and last rows $\endgroup$
    – unknown
    May 4, 2023 at 2:38
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    $\begingroup$ @unknown that $ [[9,1,3]] $ code is not $ GF(4) $ linear. It is not an even code since it has weight $ 1 $ stabilizer generators like $ IIIIIXIII $. All $ GF(4) $ linear codes are even codes by theorem 4 of arxiv.org/pdf/quant-ph/9608006.pdf $\endgroup$ May 4, 2023 at 13:24
  • $\begingroup$ right...a few things didn't seem right about the matrix; that's why I checked for GF(4) linearity...i missed more obvious signs of a problem. You can contact the author if you like; it could be a typo of some sort. $\endgroup$
    – unknown
    May 4, 2023 at 15:56

1 Answer 1

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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]]

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

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