All stabilizer codes and also all non stabilizer codes that I am aware of—for example the ones here—have a basis of codewords which are all uniform modulus superpositions of computational basis kets, in the sense that every nonzero coefficient has modulus $\frac1{\sqrt{|S|}}$ (where $ |S| $ is the size of the support of the codeword). In the particular case of stabilizer codes every nonzero coefficient has modulus $ \frac1{\sqrt{2^r}} $ for some fixed $ r \leq n-k $; for a reference see here.
What is an example of a (necessarily non-stabilizer) code for which the code space is not spanned by codewords which are all uniform modulus superpositions? So to reiterate I'm looking for a $ d=2 $ code for which the code space is not spanned by codewords which are all uniform modulus superpositions.
