# Does a $[[5,1,2]]$ CSS code exist?

Every $$[[5,1,3]]$$ code is equivalent to the perfect five qubit code with stabilizer generators $$XZZXI \\ IXZZX \\ XIXZZ \\ ZXIXZ$$ see for example

Equivalence between Quantum Error Correcting codes and uniqueness of the $[\![5,1,3]\!]$ code

The $$[[5,1,3]]$$ code is not equivalent to any CSS code. See

CSS Code in disguise

So no $$[[5,1,3]]$$ CSS code exists.

My question is:

Does a $$[[5,1,2]]$$ CSS code exist?

• Just curious, why not take the $[[4,1,2]]$ and extend it by a single fixed qubit? You can always make codes 'worse' in this way. Mar 24, 2023 at 15:50
• @squiggles ya I thought about that and I realized I actually don't know how to just take a CSS code and extend it to a CSS code on one more quibt! What is the stabilizer for a $[[5,1,2]]$ CSS code extended from a $[[4,1,2]]$ CSS code? Mar 24, 2023 at 15:58
• e.g. for the $[[4,1,2]]$ given by $\langle ZZZZ, XXXX, ZZII\rangle$, you can make the $[[5,1,2]]$ code that is just appending a fixed '0' qubit $\langle ZZZZI, XXXXI, ZZIII, IIIIZ \rangle$. Mar 24, 2023 at 18:45

$$S=\langle ZZZZZ, XXXXI, IXXXX, XXIXX\rangle$$ with logicals $$L_X = XXIII, L_Z= IZIZI$$ should do the trick.
• oh nice this even gives a $[[5,2,2]]$ CSS code $S=\langle ZZZZZ, XXXXI, IXXXX \rangle$ and in general gives an $[[2n+1,2n-2,2]]$ CSS code Mar 24, 2023 at 12:27
• You can also obtain a $[\![5,2,2]\!]$ CSS code by extending the $[\![4,2,2]\!]$ (CSS) code $S = \langle XXXX, ZZZZ \rangle$ by a single fixed qubit. Aug 17, 2023 at 14:38
A $$[[5,1,2]]$$ code occurs as a member of a family of hypergraph product codes with parameters $$[[2d^2-2d+1,1,d]]$$ codes : $$[[5,1,2]],[[13,1,3]],[[25,1,4]],\cdots$$. These are all CSS codes. Here are (quantum) tanner graphs of the first three :