[[12,3,3]] — weight-4 non-CSS stabilizer code

Direction & hypothesis

Among the general-stabilizer board's submissions with generator weight <= 4, the best kd^2/n was 1.999 when this entry was prepared (codes/761-1-39.json, @mrvee-qC-bee); the literature baseline codes/8-2-3.json (Cross and Vandeth) reaches 2.25. Weight <= 4 is the leaderboard's weight slider, not a separate track. Wei, Han, He, Li and Liu (arXiv:2601.19848) tabulate the lowest possible generator weight of small stabilizer codes; their Table 1 lists a weight-4 [[8,2,3]] code of Cross and Vandeth (arXiv:2501.17447), at kd^2/n = 2.25, and their Table 2 lists the lowest weight for [[12,3,3]] as 4-5. The hypothesis: a direct SAT search over weight-4 generator sets on 12 qubits decides whether weight 4 is enough for [[12,3,3]], which would give kd^2/n = 2.25 with k = 3.

Also builds on: the small-code weight tables of Wei, Han, He, Li and Liu (arXiv:2601.19848) and the weight-4 [[8,2,3]] code of Cross and Vandeth (arXiv:2501.17447).

Board precedents: the Cross and Vandeth [[8,2,3]] is on the board as a literature baseline (codes/8-2-3.json), with the same kd^2/n = 2.25; this entry has one more logical qubit and is not dominated by it. Weight-4 non-CSS stabilizer codes are already on the board from @mrvee-qC-bee (codes/761-1-39.json), @FarLab (codes/25-1-7.json) and @MathysRennela (codes/14-3-3.json, codes/13-1-5.json). SAT-based discovery of small codes is already on the board from @MathysRennela. The board's codes/12-3-3.json (@vprusso) is a different, CSS code with generators of weight up to 6; this one is non-CSS, hence the -s slug, as in @FarLab's codes/18-2-5-s.json.

Our contribution: the SAT search that found these generators, and the exact distance and non-CSS checks below.

Frontier: Dominates codes/14-3-3.json ([[14,3,3]], weight 4, @MathysRennela) on n; not dominated by any entry in the weight-4 cell. It does not lead the board's higher-weight stabilizer cells.

What was searched

• Unknowns: the Pauli letters of 9 generators on 12 qubits, each of weight <= 4. Every pair of generators commutes (exact XOR constraints), and the 36 single-qubit Paulis have distinct nonzero syndromes, so every Pauli of weight 1 or 2 is detected (d >= 3, no stabilizer element of weight <= 2). One symmetry-breaking clause: generator 0 contains qubit 0. Solved with kissat in 1.9 s on one thread.
• The same probe at other sizes: [[13,3,3]] (10 generators) was found too (kd^2/n 2.08, lower); [[10,3,3]] (7 generators) was undecided after 600 s; a later complete encoding with a local-Clifford normal form found no weight-4 [[10,3,3]] without a weight-2 stabilizer element.
• Group-invariant searches (qubits in orbits of a small abelian group, distance imposed lazily) found weight-4 codes at the same ratio, [[16,4,3]], [[24,6,3]] and [[32,8,3]]; at rate 1/3 and d = 3 ([[12,4,3]] under Z4 and Z2 x Z2, [[15,5,3]] under Z5, [[18,6,3]] under Z6) they found no code of that shape.

Evidence trail

Exact distance for these generators: an exhaustive enumeration of every Pauli of weight 1 and 2 (630 operators) finds no nontrivial logical and no stabilizer element; at weight 3 there are 36 nontrivial logicals, among them the stored witness (X on 0, 1 and Z on 0, 2: Y0 X1 Z2). A second, separate exhaustive program gives the same d = 3.

| effort (CPU-hours) | lightest logical found | |---|---| | 0.01 (exhaustive over all Paulis of weight <= 2, then weight 3) | 3 (no logical of weight 1 or 2) |

Claim: d = 3, filed as a witness-backed upper bound because the board's Pauli-weight certifier is not available yet; the exhaustive enumeration shows it is exact. No trial counts: the distance check is exhaustive, not random.

Non-CSS: the verifier finds no single-qubit Clifford frame that makes the generators CSS, and two separate tests (a search over the group elements and a linear-algebra test) find none for the whole stabilizer group. k = 3 from the rank of the 9 independent generators.

Literature: Grassl's codetables.de gives d = 4 as the optimum for [[12,3]] without a weight limit, so this code is not distance-optimal; its point is the generator weight. Wei, Han, He, Li and Liu (arXiv:2601.19848, Table 2) list the lowest weight for [[12,3,3]] as 4-5, with a construction of weight 5. We did not find a weight-4 [[12,3,3]] stabilizer code there, in arXiv:2601.15446 (Wang, Liu, Li, Kubica and Gu) or in the EC Zoo (errorcorrectionzoo.org); literature novelty is not claimed.

Dead ends

• Rate 1/3 at d = 3 under small groups (see above): no code of these shapes.
• d = 4 at weight 4 above kd^2/n = 2: rate 1/3 under Z4, Z5, Z6 and Z2 x Z2 gave no code of that shape; [[11,2,4]] and [[12,2,4]] without symmetry were undecided after 15 min each.
• The same ratio-2.25 shape at rate 1/4 and d = 4 (every qubit in exactly 3 weight-4 generators, under Z4 to Z8): undecided; no such code was returned.

Tools

Claude Opus 5.5 (Claude Code) working for @MicheldeMessieres. Python with PySAT for the encoding, kissat as the solver, numpy and plain Python enumeration for the distance and non-CSS checks. Scripts are local (not in this tree); the method is described above. Checked with verify/qldpc_verify.py and verify/validate_candidate.py. Compute: under 0.01 CPU-hours for the find and the distance check; about 6 CPU-hours for the whole weight-4 exploration including the dead ends.

Reproduction

Qubits 0-11, one generator per line (letters for qubits 0 to 11):

• YZXIXIIIIIII
• YIIIIIIIXYIX
• IXIIZIZXIIII
• XYIIIIIYIIIY
• IIZYZIIIIYII
• IIIIIZXYIIZI
• IIIZXXYIIIII
• IIIIIXIIZIYY
• IIIXIZIIYXII

These are exactly the entry's checks.S rows 0-8.