# How many $[[9,1,3]]$ surface codes are there? Do they include all $[[9,1,3]]$ CSS codes?

Two codes are said to be equivalent if their code spaces are related by a non-entangling gate, i.e., a gate from $$U(2)^{\otimes n} \rtimes S_n$$, the local unitaries together with permutations.

The paper Projective Plane and Planar Quantum Codes lists three non-equivalent $$[[9,1,3]]$$ CSS codes coming from different cellulations of the projective plane (Figures 2, 3 and 4). The code from Figure 4 is the Shor code.

In the answer here https://quantumcomputing.stackexchange.com/a/27699/19675 Adam Zalcman lists yet another $$[[9,1,3]]$$ surface code with explicit stabilizer generators given in the comments.

All four of these codes are non-equivalent. The codes from figure 2,3 are both odd codes but the code in figure 3 has fewer $$Z$$ type stabilizer generators (because there are fewer vertices in the cellulation, in general the number if $$Z$$ type stabilizer generators is the number of vertices minus $$1$$). Similarly, the Shor code and the code described by Adam Zalcman are both even but the Shor code given in figure 4 has fewer $$Z$$ type generators.

I'm curious how many different $$[[9,1,3]]$$ surface codes there are. Is it just these four? Or are there other $$[[9,1,3]]$$ surface codes not equivalent to any of these four? Is there some way to count the number of inequivalent $$[[9,1,3]]$$ surface codes? Is it possible that all $$[[9,1,3]]$$ CSS codes are equivalent to some surface code?

Not sure if this is too trivial, but take the $$[[7,1,3]]$$ code and add two dummy qubits.
• I see. Could you explain more about why adding two dummy qubits to the $[[7,1,3]]$ code cannot be a surface code? That claim seems plausible but I'm just not very familiar with techniques for showing that a given CSS code cannot be any surface code. Even if you don't have a totally formal proof, I would just be curious to hear any partial proof/approaches you might suggest to showing that the [[7,1,3]] code plus 2 dummy qubits is not a surface code. If you think it's better I can make a separate question "Proof that Steane 7 qubit code is not equivalent to any surface code" Jun 15 at 15:41