# How many $[[9,1,3]]$ surface codes are there?

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 his answer to Different codes with the same parameters 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 of $$Z$$ 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?

• Not answering your good question, but FWIW [[7,1,3]] codes have been classified; here is on that is a surface code (diff from Steane) errorcorrectionzoo.org/c/twist_defect_surface Commented Jul 2 at 14:23