\(((8,16,2))\) \(PG(3,2)\) 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 and Permutation-Based 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 \(2\ket{\overline{0}}\bra{\overline{0}}-I\) 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].
Primary Hierarchy
Parents
\(((8,16,2))\) \(PG(3,2)\) code
References
- [1]
- A. S. Morris and D. Malz, “Constraints on phantom codes from automorphism group bounds”, (2026) arXiv:2604.15111
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2026-05-12)
Cite as:
“\(((8,16,2))\) \(PG(3,2)\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qubit_8_4_2, arXiv:2606.11484