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.
This isolated discrepancy stems from the exceptional isomorphismGL 4(F2) ∼= A8, which implies thatµ(GL 4(F2)) = 8 in contrast to the usual value µ(GLk(F2)) = 2 k −1
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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.