The quantum repetition code is an $ [[n,1,1]] $ stabilizer code with stabilizer generators $ Z_iZ_{i+1} $ for $ i=1, \dots, n-1 $.
The Eastin-Knill theorem states that a $ d >1 $ code cannot have a universal set of transversal gates, in fact it can only have finitely many transversal gates.
However, a quantum repetition code has distance $ d=1 $ so the Eastin-Knill theorem does not restrict the transversal gates.
Indeed any diagonal gate $$ \begin{bmatrix} e^{i \phi} & 0 \\ 0 & e^{i \theta} \end{bmatrix} $$ can be implemented transversally on a quantum repetition code.
Thus the $ [[n,1,1]] $ quantum repetition code is a good example of a $ d=1 $ code with infinitely many transversal gates.
Also the Pauli $ X $ gate is exactly transversal, meaning that $ X^{\otimes n} $ implements logical $ X $.
The $ X $ gate together with the diagonal gates forms the group of all single qubit gates that do not create superpositions, a large subgroup of $ U(2) $. But it turns out that no other transversal gates exist for this code i.e. there is not a universal set of transversal gates, see Does the Eastin-Knill theorem hold for repetition codes? . Essentially the argument is that a transversal logical gate is, by definition, implemented by a physical gate of the form $$ U_1\otimes U_2\otimes \dots \otimes U_n $$ i.e. a non entangling gate. So if some codewords in the codespace are unentangled then it will be impossible to use transversal logical gates to create a superposition and thus a universal set of transversal gates is ruled out.
However this argument hinges on a highly unusual property of the repetition code: there are codewords which are unentangled.
For all $ d>1 $ code that I am aware of, every codeword corresponds to a state in which all qubits are entangled with all other qubits.
My question is, what is an example of a $ d=1 $ code with a universal set of transversal gates?
There is of course a silly example which is the $ [[1,1,1]] $ code with encoding map $$ |\psi \rangle \mapsto |\psi \rangle $$ or its generalization the $ [[n,1,1]] $ code with stabilizer generators $ Z_i $, $ i=2,\dots n $ and encoding map $$ |\psi \rangle \mapsto |\psi \rangle \otimes | 0 \rangle^{\otimes (n-1)} $$
Or is every code with a universal set of transversal gates trivial?
