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?

Note that not all $ [[9,1,3]] $ CSS codes are surface codes as you can add two dummy qubits to the Steane code, and the Steane code is generally thought to not be a surface code on any surface.

Is the [[7,1,3]] Steane code a surface code?

Is every $ [[9,1,3]] $ CSS code a surface code?



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