There is a famous $ [[3,1,2]]_3 $ qutrit stabilizer code with stabilizer generators $$ XXX $$ $$ ZZZ $$ where it should be clear from context that $ X $ and $ Z $ here denote the appropriate qudit versions $$ X= \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} $$ and $$ Z= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \omega & 0 \\ 0 & 0 & \omega^2 \end{bmatrix} $$ where $ \omega=e^{2 \pi i /3} $. Similarly for any qudit of dimension $ q $, with $ q $ odd, there is a $ [[3,1,2]]_q $ stabilizer code with stabilizer generators given by $$ XXX $$ $$ ZZZ^{-2} $$ where $ X $ and $ Z $ here are the appropriate $ q \times q $ qudit Pauli matrices (i.e. the "clock and shift" matrices given here https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices ). Also to be clear $ Z^{-2} $ is just the square of the inverse and could also be written as $ Z^{-2} =(Z^2)^\dagger=(Z^\dagger)^2 $ (note that for a qutrit $ Z^{-2}=Z $ so this construciton just yields the standard 3 qutrit code).
What about qudits of dimension $ q $ with $ q $ even? Then this construction fails since the code cannot detect the single qudit error $ IIX^{q/2} $ and so the distance will be $ d=1 $ instead of $ d=2 $. And at least for qubits ( $q=2$ ) we know it is impossible to have a $ [[3,1,2]] $ stabilizer code. In fact it is impossible to have any, even non-stabilizer, code encoding a single logical qubit into three physical qubits, see Why can't there be an error detecting code with fewer than 4 qubits? .
So my question becomes: For a qudit of even dimension $ q >2 $ is it always possible to have a distance $ 2 $ code encoding one logical qudit into $ 3 $ physical qudits? Is it never possible? I am also interested in the much narrower stabilizer code version of this question i.e. do $ [[3,1,2]]_q $ stabilizer codes exist for $ q>2 $ even?
