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.
It is proven in Corollary 10 of Quantum Codes of Minimum Distance Two that every $ ((5,2,3)) $ code is equivalent to the $ [[5,1,3]] $ stabilizer code. Note that the parameters $ ((5,2,3)) $ are extremal.
The stabilizers for an $ [[11,1,5]] $ code are given here http://www.codetables.de/QECC.php?q=4&n=11&k=1.
Note that the parameters $ ((11,2,5)) $ are also extremal. This is stated, for example, in Quantum Error Correction via Codes over GF(4)
Is every stabilizer code with parameters $ [[11,1,5]] $ equivalent to the $ [[11,1,5]] $ stabilizer code linked above?
Moreover, is every $ ((11,2,5)) $ code equivalent to the $ [[11,1,5]] $ stabilizer code? This seems plausible to me since $ ((11,2,5)) $ are extremal code parameters and we know uniqueness holds for the extremal code parameters $ ((5,2,3)) $.
