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From this site and this post, doubly even codes have transversal $S$.

Based on this post, surface codes don't have transversal $S$ gates. We can check the boundary stabilizers of surface codes are not doubly even, so surface codes are generally not doubly even and don't have transversal $S$.

However, what if we go to the toric codes which seems to be doubly even? What's the logical effect of transversal $S$ gates?

Yunzhe
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The toric code is not in general doubly even. Take the $[[18,2,3]]$ code for example; $H_X$ has this weight distribution :

[ [ 0, 1 ], [ 4, 9 ], [ 6, 24 ], [ 8, 99 ], [ 10, 72 ], [ 12, 51 ] ]

1 element of weight 0, 9 elements of weight 4, 24 elements of weight 6,....

Since there are weights that are not multiples of 4, the code is not doubly even.

unknown
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