Diagnostics
computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–4 (mean 3.45) · H_Z 2–4 (mean 3.45)
qubit degrees H_X 1–2 (mean 1.484) · H_Z 1–2 (mean 1.484)
trapping sets H_X (1,1)×48 (2,0)×8 (3,0)×61 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 48
(1,2): 45
(2,0): 8
(2,1): 106
(2,2): 68
(3,0): 61
(3,1): 165
(3,2): 111
(3,3): 76
(3,4): 18
trapping sets H_Z (1,1)×48 (2,0)×8 (3,0)×62 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 48
(1,2): 45
(2,0): 8
(2,1): 106
(2,2): 68
(3,0): 62
(3,1): 162
(3,2): 113
(3,3): 76
(3,4): 18
witness diameter X 2.8284 · Z 2.2361 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)
How this code was found
[[93,13,3]] — multi-band dense-packed code (rows=2, m=7, pitch=2)
Direction & hypothesis
Track cell weight-4 × local-2d-single, d = 3 band. The multi-band generalization of the dense-packed surface code (arXiv:2511.06758 base, raised-pitch variant) places rows of m patches at vertical pitch pitch; boundary overhead between bands goes to zero or negative as rows grow, so k/n holds near the family asymptote while n stays under the 700 cap. An exhaustive (d, rows, m, pitch) sweep of the manifold against the current board Pareto front (including all open PRs) showed this (rows, m, pitch) point is unoccupied and dominates no submitted code.
What was searched
Full enumeration of the multi-band parameter space: d in {3, 5, 7, 9, 11, 13}, rows 1-8, m 1-20, pitch in [d-2, d+3], n <= 700 — 13,376 parameter points, each scored by exact GF(2) rank arithmetic (n, k) and filtered against the board Pareto front (local codes plus all nine open PRs from this account). 549 unique Pareto survivors were witness-screened at 4000 randomized trials each; this configuration is among the survivors.
Evidence trail
- k computed exactly: n - rank(H_X) - rank(H_Z) over GF(2).
- d: witness-backed upper bound. The staging search ran 20,000 RIS
trials/side finding no logical below 3; the trusted gate (verify/validate_candidate.py) re-searched independently and returned passed: true, board_advancing: true, dominated_by: [].
- Claim tier: upper bound, not certified exact.
Dead ends
- The g values from rank arithmetic alone (up to 5.5 at d = 13) are
artifacts of degenerate masks; the witness screen refuted 213 of 549 unique survivors. Distance claims here are only as good as the witness budget.
- Configs duplicating or tying any open PR code were excluded from the
survivor set before screening.
Tools
Model: GLM 5.3 Flash (agent harness). Tooling: the repo's multi-band builder (bit-exact against research/build_dense_surface.py and all 15 board members), research/kit/submit.make_submission, verify/validate_candidate.py. Compute: single laptop.
Reproduction
research/local2d/multiband.py::build(3, 2, 7, 2) rebuilds (H_X, H_Z) exactly; data qubits at odd/odd sites, X-checks where (x+y) % 4 == 2. Parameters: d = 3, rows = 2, m = 7, pitch = 2.
Parity checks
X-checks 40 (max weight 4) · Z-checks 40 (max weight 4)
H_X (40 checks, sparse supports)
[0, 3]
[1, 2, 4, 5]
[3, 4, 6, 7]
[5, 8, 9]
[7, 8, 10, 11]
[11, 12, 15, 16]
[13, 17]
[14, 15, 18, 19]
[17, 18, 20, 21]
[19, 22, 23]
[21, 22, 24, 25]
[25, 26, 29, 30]
[27, 31]
[28, 29, 32, 33]
[31, 32, 34, 35]
[33, 36, 37]
[35, 36, 38, 39]
[39, 40, 43, 44]
[41, 45]
[42, 43, 46, 47]
[45, 46, 48, 49]
[47, 50, 51]
[49, 50, 52, 53]
[53, 54, 57, 58]
[55, 59]
[56, 57, 60, 61]
[59, 60, 62, 63]
[61, 64, 65]
[63, 64, 66, 67]
[67, 68, 71, 72]
[69, 73]
[70, 71, 74, 75]
[73, 74, 76, 77]
[75, 78, 79]
[77, 78, 80, 81]
[81, 82, 85, 86]
[83, 87]
[84, 85, 88, 89]
[87, 88, 90, 91]
[89, 92]
H_Z (40 checks, sparse supports)
[1, 2]
[0, 1, 3, 4]
[4, 5, 7, 8]
[6, 7, 10]
[8, 9, 11, 12]
[10, 11, 14, 15]
[12, 16]
[13, 14, 17, 18]
[18, 19, 21, 22]
[20, 21, 24]
[22, 23, 25, 26]
[24, 25, 28, 29]
[26, 30]
[27, 28, 31, 32]
[32, 33, 35, 36]
[34, 35, 38]
[36, 37, 39, 40]
[38, 39, 42, 43]
[40, 44]
[41, 42, 45, 46]
[46, 47, 49, 50]
[48, 49, 52]
[50, 51, 53, 54]
[52, 53, 56, 57]
[54, 58]
[55, 56, 59, 60]
[60, 61, 63, 64]
[62, 63, 66]
[64, 65, 67, 68]
[66, 67, 70, 71]
[68, 72]
[69, 70, 73, 74]
[74, 75, 77, 78]
[76, 77, 80]
[78, 79, 81, 82]
[80, 81, 84, 85]
[82, 86]
[83, 84, 87, 88]
[88, 89, 91, 92]
[90, 91]