Quantum codes: stabilizer [[n,k,d]] certificates

id: vfr_001f148c07eebecb
license: CC-BY-4.0
findings: 5
accepted core: 0
contested: 0
links: 0
sources: 5
evidence: 5
avg conf: 0.88

used by 0 · replayed by 1 producer · second seat open

e6/6 · finding.asserted · reviewer:will-blair · 2026-06-02 · null→56c6

Brief & export

findings 5 · accepted 0 · open questions 1 · contested 0

strongest · none formally accepted

finding	confidence	source
The [[4,2,2]] code with stabilizers XXXX, ZZZZ encodes 2 logical qubits in 4 physical qubits with distance 2, verified by exact recomputation of k and d.	0.97	verifier
The Steane code [[7,1,3]] (CSS from the [7,4,3] Hamming code) encodes 1 logical qubit in 7 physical qubits with distance 3, verified from its six stabilizer generators.	0.97	verifier
The perfect five-qubit code [[5,1,3]] (cyclic stabilizers XZZXI and rotations) encodes 1 logical qubit in 5 physical qubits with distance 3, and is optimal at its length.	0.97	verifier
The Shor code [[9,1,3]] (concatenated bit- and phase-flip repetition) encodes 1 logical qubit in 9 physical qubits with distance 3, verified from its eight stabilizer generators.	0.97	verifier
The largest distance of a single-logical-qubit stabilizer code at length 10, [[10,1,d]], is open within the band d in [3,4]; whether a [[10,1,4]] stabilizer code exists is undetermined here.	0.50	verifier

bibliography · 5

cap_a1b2c3d4e5f60718 · vc_qcopen000000001
cap_a1b2c3d4e5f60718 · vc_qc9130000000001
cap_a1b2c3d4e5f60718 · vc_qc5130000000001
cap_a1b2c3d4e5f60718 · vc_qc7130000000001
cap_a1b2c3d4e5f60718 · vc_qc4220000000001

export

# Quantum codes: stabilizer [[n,k,d]] certificates

This frontier holds 5 findings (0 accepted) over 5 sources.

## Significance

- The [[4,2,2]] code with stabilizers XXXX, ZZZZ encodes 2 logical qubits in 4 physical qubits with distance 2, verified by exact recomputation of k and d. (verifier)
- The Steane code [[7,1,3]] (CSS from the [7,4,3] Hamming code) encodes 1 logical qubit in 7 physical qubits with distance 3, verified from its six stabilizer generators. (verifier)
- The perfect five-qubit code [[5,1,3]] (cyclic stabilizers XZZXI and rotations) encodes 1 logical qubit in 5 physical qubits with distance 3, and is optimal at its length. (verifier)
- The Shor code [[9,1,3]] (concatenated bit- and phase-flip repetition) encodes 1 logical qubit in 9 physical qubits with distance 3, verified from its eight stabilizer generators. (verifier)
- The largest distance of a single-logical-qubit stabilizer code at length 10, [[10,1,d]], is open within the band d in [3,4]; whether a [[10,1,4]] stabilizer code exists is undetermined here. (verifier)

## Open questions

- The largest distance of a single-logical-qubit stabilizer code at length 10, [[10,1,d]], is open within the band d in [3,4]; whether a [[10,1,4]] stabilizer code exists is undetermined here.

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