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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.

Ian Gershon Teixeira
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Since we’re leaving stabilizer codes behind, let’s go even farther away: spin codes are one example.

squiggles
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  • 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". – squiggles Feb 16 '23 at 02:15
  • 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 $. – Ian Gershon Teixeira Feb 16 '23 at 03:19
  • related papers explicitly demonstrating spin codes corresponding to multiqubit codes with nontrivial distance: https://arxiv.org/abs/2304.08611 and https://arxiv.org/abs/2305.07023 – Ian Gershon Teixeira Jul 26 '23 at 21:31