\(((8,16,2))\) \(PG(3,2)\) code

\(((8,16,2))\) \(PG(3,2)\) code[1]

Description

Eight-qubit code encoding four logical qubits whose logical basis consists of a GHZ state together with fifteen states built from the incidence geometry of the projective space \(PG(3,2)\).

Protection

Distance two, and this is optimal because no \(((8,16,3))\) code exists [1]. No Pauli stabilizer subsystem phantom code of type \([[8,4,r,d\geq2]]\) exists, so this exceptional \(k=4\) phantom code is necessarily nonstabilizer [1].

Transversal Gates

Even physical-qubit permutations act as \(GL(4,\mathbb{F}_2)\) on the logical basis, and odd permutations extend the permutation automorphism group to the full symmetric group \(S_8\) [1].\(T^{\otimes 8}\) is a transversal non-Clifford gate implementing \(2\ketbra{\overline{0}}-\mathbbm{1}\) on the logical subspace [1].A specific odd permutation implements a non-Clifford logical involution, and the full permutation automorphism group is \(S_8\) [1].

Cousins

Phantom code— This is the exceptional nonstabilizer \(k=4\) qubit phantom code of minimal length eight that violates the generic bound \(n\geq 2^k-1\) [1].
Self-complementary qubit code— The logical basis of the \(((8,16,2))\) \(PG(3,2)\) code contains a GHZ state and linear combinations of self-complementary states [1].