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

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?

• 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 May 4 at 2:38
• @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 May 4 at 13:24
• 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. May 4 at 15:56

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]]


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