# $[[n,1,d]]$ codes with $n$ even

All the interesting/well-known $$[[n,1,d]]$$ codes I know have $$n$$ odd. Moreover, all these codes have the property that $$X^{\otimes n}$$ implements logical $$X$$ and $$Z^{\otimes n}$$ implements logical $$Z$$ (this property of course implies that $$n$$ is odd since logical $$X$$ and logical $$Z$$ must anticommute).

Are there any interesting/well-known $$[[n,1,d]]$$ codes for $$n$$ even? If so what are logical $$X$$ and logical $$Z$$ for those codes?

Note: I don't consider the $$[[4,1,2]]$$ code given by adding another stabilizer like $$ZZII$$ to $$$$ to be an interesting code.

• what about the Toric code? Nov 8, 2022 at 8:58
• toric code has k=2 Nov 9, 2022 at 16:38
• @DaftWullie I guess by Toric code you mean some kind of $[[n,1,d]]$ surface code? Nov 9, 2022 at 16:40
• look here codetables.de for many examples, For example $[[6,1,3]]$, $[[8,1,3]]$,... Nov 9, 2022 at 16:44
• @unknown ok this is a very cool reference. I'm happy with the $[[24,1,8]]$ code given there as an answer. Interesting to note that for every even number less than 24 the $[[n,1,d]]$ code in the table is explicitly described as either an extension of a well known $[[n,1,d]]$ code for $n$ odd or as a restriction of a well known $[[n,2,d]]$ code for $n$ even. Anyway I would definitely consider that $[[24,1,8]]$ code as an example of an $[[n,1,d]]$ code of independent interest. Now I'm just curious what the logical $X$ and logical $Z$ are for that code? Nov 9, 2022 at 16:54

Here's the $$[[24,1,8]]$$ code and its logicals :

stabilizers : [

[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0],
[0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0],
[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,1,1,1,1,0,1,1,1,1,1,1,0,1],
[0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0,0,1,0,1,1,1,1,1,1,0,0,1],
[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,0,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0,1,1,1,1,0,1,1],
[0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,1,0,0,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0],
[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,1,0,0,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1],
[0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0,1,0,0,1],
[0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,1,0,0,0,0,1,1,1,0,1,1,1],
[0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,1,0,0,1,1,1,0,0,1,1,0,1,1,0,0,0],
[0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,1,0,0,1,1,1,0,0,1,1,0,1,1,0,0],
[0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,1,0,0,1,1,1,0,0,1,1,0,1,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,1,1,1,1,1,1,1,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,1,1,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,0,0,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,1,0,0,0,0,1,1,0,0,1,0,0,0,0,1,1,0,0,0,1,0,1,1,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0,0,0,1,0,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1]]


logicals: [

[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,1,1,1,0,1]]


I checked that its distance is 8.

• Interesting, it seems that $Z^{\otimes 24}$ and $X^{\otimes 24}$ both commute with the stabilizer so they both implement logical operations. I'm guessing neither of them are in the stabilizer in which case they would be implementing nontrivial logical operations. Since logical $X$ and logical $Z$ anticommute it can't both be the case that $Z^{\otimes 24}$ implements logical $Z$ and $X^{\otimes 24}$ implemenets logical $X$. I wonder what logical operations they actually implement(or maybe they are in the stabilizer and I just can't see it)? Nov 9, 2022 at 21:06
• Try adding the last 4 stabilizers together (found by using mathematica. I set the stabilizer matrix above to a variable code and executed LinearSolve[Transpose[code], Join[Table[0, {24}], Table[1, {24}]], Modulus -> 2] Nov 10, 2022 at 8:00