Entry facet
Representation ladder
R1 checks through R5 cross-model views, with canonicality and provenance.
- R1
Checks
Sparse Pauli generators in symplectic form, including stabilizer and gauge rows.
symplectic-css-checks derived by quantum_hamming_15_7_3.build -> CheckMatrix.from_css v1
Duplicating the four binary Hamming rows into X and Z sectors gives eight commuting symplectic generators. This executable matrix fixes qubit and syndrome ordering for rank, centralizer, and exact-distance calculations.
References: steane1996simple, calderbank1996good
stabilizer-containment derived by constructor.reed_muller_gauge_out_witness + qeclib.gf2.rank v1
The [[15,7,3]] stabilizer is a strict subgroup of the repository's [[15,1,3]] quantum Reed–Muller stabilizer in the shared nonzero-vector qubit order. Their four X checks agree. The Hamming Z row space sits inside the Reed–Muller Z row space, whose six additional independent rows fix six Hamming logical Z values and leave one protected qubit.
References: steane1996simple, steane1996quantumrm
- R2
Structure
CSS pairs, chain complexes, lattices, and other structure behind the checks.
hamming-column-css canonical
Order the columns as 0001, 0010, ..., 1111. Every nonzero four-bit syndrome is therefore the address of exactly one physical qubit. The four coordinate rows are mutually orthogonal over F_2, so duplicating them as X and Z checks produces a commuting CSS presentation.
References: steane1996simple, calderbank1996good
- R3
Dynamics
Ordered measurement schedules and their instantaneous stabilizer groups.
No representation at this rung is recorded for this entry.
- R4
Circuit & detectors
Named syndrome-extraction circuits and detector error models.
No representation at this rung is recorded for this entry.
- R5
Cross-model views
Executable or illuminating algebraic, ZX, fermionic, tensor, and statistical dictionaries.
syndrome-address derived by constructor.HAMMING_CHECKS column lookup v1
A single X error anticommutes with the Z checks in the column named by its location, and a single Z error gives the same address in the X syndrome. A Y error activates both sectors. Thus the classical Hamming lookup rule survives intact, once for each Pauli component.
References: gottesman1997stabilizer
finite-geometry-automorphisms curated from grassl2013automorphisms
GL(4,2) acts on the fifteen nonzero column labels. Each invertible linear relabeling preserves the Hamming row space, hence gives a physical coordinate permutation that preserves the stabilizer. Its logical action must still be reduced in the fourteen-dimensional logical symplectic quotient; it is not merely a relabeling of seven named logical qubits.
References: grassl2013automorphisms
hamming-bound-audit derived by instance claims + single-error sphere count v1
The word “Hamming” refers to the inherited classical parity-check construction, not saturation of the nondegenerate quantum Hamming bound. The constructor-derived n, k, and distance leave unused syndrome space after all single-qubit Pauli errors are counted.
References: gottesman1997stabilizer