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The arguments leading to the Quantum Hamming Bound for non-degenerate $[[n,k,d = 2t+1]]$ codes $$\sum_{j=0}^t 3^j \binom{n}{j} ≤ 2^{n-k}$$ can be manipulated to find a stronger related bound for non-degenerate CSS codes, $$\left(\sum_{j=0}^t \binom{n}{j}\right)^2 ≤ 2^{n-k}.$$ Note that this is simply the classical Hamming bound with the left-hand side squared.

Examples:

  • the quantum Hamming codes $[[2^r-1, 2^r - 2r - 1, 3]]$ fall short of the normal QHB but would saturate this one, as LHS = RHS = $2^{2r}$,
  • Shor $[[9,1,3]]$ code has a strict inequality $100 < 256$,
  • Gottesman $[[2^r, 2^r - r - 2, 3]]$ codes violate it but they aren't CSS,
  • surface codes violate it but they are degenerate. (example disputed in comments)

I think this is pretty straightforward, but I can't seem to find it anywhere to cite it. Does it have a name? Or perhaps it's wrong, can you help me find a counterexample? Or did I just discover something?

The Vee
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This seems to be similar to what is discussed here, which is the only place I've seen perfect CSS codes defined: The [[15,7,3]] code is "only perfect as a CSS code, i.e., the 2^4 Z stabilizer syndromes are exactly enough to correct the 1 + 15 possible trivial or weight-one X errors, and similarly for Z errors."

Victor V Albert
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