# Codes with codewords that aren't uniform modulus superposition

All stabilizer codes and also all non stabilizer codes that I am aware of, for example the ones here,

Example non-stabilizer code?

have a basis of codewords which are all uniform modulus superpositions of computational basis kets, in the sense that every nonzero coefficient has modulus $$1/\sqrt{|S|}$$ where $$|S|$$ is the size of the support of the codeword.

In the particular case of stabilizer codes every nonzero coefficient has modulus $$1/\sqrt{2^r}$$ for some fixed $$r \leq n-k$$. For a reference see https://quantumcomputing.stackexchange.com/a/27573/19675

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.

• Do non-uniform superpositions of (stabilizer) codewords count? Because then we can go back to stabilizer codes. To be a bit cheeky, a single unencoded qubit is a stabilizer code (with stabilizer group $\{I\}$) but clearly has code states that aren't "uniform". Commented Feb 16, 2023 at 2:15
• Sure but the code space is still spanned by uniform codewords because you can pick $|0>$ and $|1>$. Also that trivial code does not have $d \geq 2$. Commented Feb 16, 2023 at 3:19