the quantum Hamming bound and the perfect five-qubit code
Theorem.the quantum Hamming bound and the perfect five-qubit code — the error-correction wave (a tied-thinnest ray, named by the research): a code correcting any single-qubit error must give each of the 3n+1 single-qubit Paulis (identity plus X, Y, Z on each of n qubits) a distinct syndrome, so a non-degenerate [[n,k]] code obeys 2^(n−k) ≥ 3n+1.
Proof.the error-correction wave (a tied-thinnest ray, named by the research): a code correcting any single-qubit error must give each of the 3n+1 single-qubit Paulis (identity plus X, Y, Z on each of n qubits) a distinct syndrome, so a non-degenerate [[n,k]] code obeys 2^(n−k) ≥ 3n+1 — the quantum Hamming (sphere-packing) bound. The [[5,1,3]] code SATURATES it: 2⁴ = 16 = 3·5+1, every syndrome used exactly once — the PERFECT, smallest code correcting an arbitrary single-qubit error, with n = 5 the minimum (n ≤ 4 cannot reach 3n+1 syndromes). The [[7,1,3]] Steane and [[9,1,3]] Shor codes satisfy the bound with room to spare; the Shor code is degenerate (256 syndromes for 28 errors). Counting the syndromes bounds how small a quantum code can be. This is the counting bound for non-degenerate codes (necessary, not sufficient); that the [[5,1,3]] code exists at distance 3 (Laflamme–Miquel–Paz–Zurek / Bennett 1996) is cited.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/2/8/index.ts#theQuantumHammingBoundAndThePerfectFiveQubitCode
1 · Classification
finite-complete — computed witness within a cited frame (the unbounded form leans on the cited literature)
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"the quantum Hamming bound and the perfect five-qubit code" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation theQuantumHammingBoundAndThePerfectFiveQubitCode.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (theQuantumHammingBoundAndThePerfectFiveQubitCode @ src/2/8) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes theQuantumHammingBoundAndThePerfectFiveQubitCode (src/2/8/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.