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

Question

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?

Answer 1

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 :

Answer 2

$S=\langle ZZZZZ, XXXXI, IXXXX, XXIXX\rangle $ with logicals $L_X = XXIII, L_Z= IZIZI$ should do the trick.