# Generators for $[[9,1,3]]$ linear quantum code

In Theorem 7 of the paper Quantum Error Correction Via Codes Over GF(4) the authors come up with a sort of reduction algorithm. Right after they claim the $$[[85,77,3]]$$ Hamming code implies the existence of a $$[[9,1,3]]$$ linear quantum code.

To be sure, linear here means $$GF(4)$$-linear, i.e., if we take the elements in the stabilizer of the code and consider them as elements in $$GF(4) = \{0,1,\omega,\omega^2\}$$ then not only are they closed under addition they are also closed under $$\omega$$ and $$\omega^2$$.

I was wondering what the stabilizer generators for this linear $$[[9,1,3]]$$ code are.

• the results in that paper are hard to verify (and possibly wrong). Take the (classical) hamming code over GF(4) : $[85,81,3]_4$; the example on page 19 claims that its dual is self-orthogonal and leads to a $[[85,77,3]]$ quantum code. According to GAP/GUAVA IsSelfOrthogonalCode(DualCode(HammingCode(4,GF(4))) is false; so I'm not sure how they get the $[[85,77,3]]$. In principle if you have that code you can construct the $[[9,1,3]]$ from it according to the recipe they give. May 4, 2023 at 18:14

(partial answer; too long to fit as a comment)

The paper and GUAVA use different definitions for GF(4) orthogonality. With the paper's definition, Hamming(4,GF(4)) is self orthogonal. This gives a $$4 \times 85$$ matrix over GF(4), $$H$$, which corresponds to a $$[[85,81,3]]$$ code; taking $$H$$ and $$\omega H$$ gives an $$8 \times 85$$ matrix $$H'$$ that corresponds to the $$[[85,77,3]]$$ code in the paper. Note that the GF(4) rank of $$H'$$ is still 4, but it defines a different code than $$H$$. I can get these matrices and verify the weight enumerator of $$C=(85,2^4)$$ and its dual $$C^\perp$$ (page 15). So the "answer" to your question is then : take this $$8 \times 85$$ matrix and delete certain 76 of the 85 columns to get an $$8 \times 9$$ matrix which defines the $$[[9,1,3]]$$ code. These 76 columns are the support of all codewords in $$C^\perp$$ of weight 76. I don't beleive that the problem of finding these columns is tractable for codes this size but I could be wrong.

[update] Instead of calculation which 76 columns to delete I did a random search of which 9 to keep. This found solutions fairly quickly (which means these are not that rate). Here's one :

[[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]]


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

• I can confirm what you wrote down is both $GF(4)$-linear and corresponds to a $[[9,1,3]]$ code. May 7, 2023 at 4:27