All the Stabilizer Codes of Distance 3
All the Stabilizer Codes of Distance 3
We give necessary and sufficient conditions for the existence of stabilizer codes <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$[[n,k,3]]$</tex> </formula> of distance 3 for qubits: <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n-k\geq \lceil \log _{2}(3n+1)\rceil +\epsilon _{n}$</tex></formula> , where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\epsilon _{n}=1$</tex></formula> if <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n=8 {{ 4^{m}-1}\over { …