Constant-excitation CSS codes over 𝔽ₚ: nine qutrits and eight ququints, against twelve qubits
Abstract
A quantum code is constant excitation (CE) if its whole code space lies in a single eigenspace of the total excitation operator N=Σⱼ diag(0,…,p-1)ⱼ; such a code acquires only a global phase under the collective coherent drift e^(-iθ N). For qubits N is the Hamming weight; there Lai, Liou and Ouyang proved that no distance-3 CE CSS code with one logical qubit exists on nine qubits or fewer and exhibited one on twelve, and twelve is now known to be the exact threshold. We ask the same question for p-level systems, which no source we could find asks, and answer it at p=3 and p=5: there is no [[n,1,d ≥ 3]]₃ constant-excitation CSS code for any n ≤ 8 and any global shift, and there is one at n=9; there is no [[n,1,d ≥ 3]]₅ one for any n ≤ 7, and there is one at n=8. So the least block length is 9 for qutrits and 8 for ququints, against 12 for qubits: the qutrit answer is smaller than the qubit one, and a table indexed only by p=2 hides that. Both witnesses are exhibited explicitly. We also prove an identity that holds for every CE CSS code over 𝔽ₚ — the shift's excitation on the support of a codeword v satisfies 2Σ_(i ∈ supp v) yᵢ = wt(v)(p-1) — whose p=2 case is the containment condition the qubit treatment takes as a definition. The distance conditions we refute are the true stabilizer distances, so degenerate codes are excluded as well, and only additive closure is assumed of C₁. The nonexistence halves rest on exhaustive searches over a space cut down by three structural theorems; both the reduction and the completeness of the enumeration are theorems, not assumptions, and every statement below is machine-checked in Lean 4.
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Source snapshot 2026-08-30 15:34 UTC
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2419c28844b56e15b03c74b0d892c34ff175ac57b255592ff426f9e1e7a2d20f
Claim ledger
Stated results
Q1routine2026-08-30
The Fₚ reduction: additive closure over a prime field is Fₚ-linearity, the distance condition is the true one, coordinate permutations transport a code, and the shift may be taken with non-increasing digits
Q2routine2026-08-30
The pruned depth-first enumeration of subspaces of V_y over Fₚ is complete, and a five-number certificate is a code
Q3candidate2026-08-30
Headline: 9 is the least n admitting an [[n,1,d>=3]]₃ constant-excitation CSS code – smaller than the qubit answer 12
Q4routine2026-08-30
Regression at p = 2: the Fₚ machinery reproduces families/ce-css exactly – no CE CSS code for n <= 11, and the source's [[12,1,3]] code
Q5routine2026-08-30
Negative controls: too strong, too weak, and the node check is not vacuously rejecting
Q6routine2026-08-30
The shift's excitation on the support of any codeword is forced: 2 sum_(i in supp v) yᵢ = wt(v)(p-1); for qubits every codeword has even weight
Q7candidate2026-08-30
8 is the least n admitting an [[n,1,d>=3]]₅ constant-excitation CSS code – completing the CE CSS thresholds 12, 9, 8 at p = 2, 3, 5, of which 9 and 8 are this family's (Q3, Q7) and 12 is families/ce-css
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- For each prime p, what is the least n admitting an [[n,1,d ≥ 3]]ₚ constant-excitation CSS code? The qubit answer is 12 (families/ce-css). The qutrit answer is 9 — smaller — and the ququint answer is 8, so the price of constant excitation is not monotone in p.
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- 2026-09-07 03:53 UTC
- Ledger commit
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