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 (relying heavily on results from the paper Polynomial Invariants of Quantum Codes by the same author) 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? This is not obvious, for example there are many non equivalent stabilizer codes with parameters $ [[9,1,3]] $ see for example How many $ [[9,1,3]] $ surface codes are there?
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)) $.
