Phantom codes obey the bound k ≤ log₂(n+1) except for a constructed (8, 2^4, 2) example at k=4, following from a general theorem linking code length to automorphism group structure.
It follows that the action ofσ c on the point-star states is σc |ax⟩= X ℓ∋x σc |ℓ⟩= X ℓ∋x ℓ⊥ = X ℓ⊥∋x |ℓ⟩= X ℓ⊂x⊥ |ℓ⟩=|p x⊥ ⟩,(S56) wherex ⊥ is the plane dual to the pointx
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Constraints on phantom codes from automorphism group bounds
Phantom codes obey the bound k ≤ log₂(n+1) except for a constructed (8, 2^4, 2) example at k=4, following from a general theorem linking code length to automorphism group structure.