← back to the board
[[96,8,12]] d =
n
96
k
8
d
12
kd²/n
12.0
w
8
X/Z
1
g
0.0088
r
6.0828
layers
2
swaps
571

Share this result

Distance

X/Z asymmetry 1 · d_X = 12, d_Z = 12 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 12 · witness weight 12 (claimed upper_bound)
witness found by @MathysRennela · research/kit/surrogate.lightest_logical · found at 2×105 trials · survived 2×105 trials · 2026-10-06
witness operator (support, 12 qubits)
[0, 10, 12, 20, 48, 49, 50, 53, 56, 59, 86, 89]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness found by @MathysRennela · research/kit/surrogate.lightest_logical · found at 2×105 trials · survived 2×105 trials · 2026-10-06
witness operator (support, 12 qubits)
[15, 16, 17, 28, 32, 33, 34, 45, 54, 60, 68, 72]
certificate exact, d = 12 · CryptoMiniSat 5.14 SAT
X: no logical < 12 exists; Z: no logical < 12 exists

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×96 (2,4)×144 (3,4)×384 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 96 (2,4): 144 (2,6): 1056 (3,4): 384 (3,6): 3552 (3,8): 16416 (3,10): 1152
trapping sets H_Z (1,4)×96 (2,4)×144 (3,4)×384 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 96 (2,4): 144 (2,6): 1056 (3,4): 384 (3,6): 3552 (3,8): 16416 (3,10): 1152
witness diameter X 8.0623 · Z 5.099 (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)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
layout contributed by @dorakingx · Simulated annealing over site assignments on an integer grid with the repo's own research/local2d/fold_layout.search_layout (max-diameter objective, several bounding boxes and seeds); single layer at radius 4 tried first, then two layers at 7. Radius and per-site occupancy re-measured with the verifier's rule. · 2026-10-07
r = 6.083
X checkZ checkqubit site (48)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 571 nearest-neighbor SWAPs per round in total, at most 11 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction Two-block group-algebra (2BGA) code C(a,b) over G = D_4 x Z_6, built exactly as specified in arXiv:2610.05307 (Appendix J, supports of the [[96,8,12]] row of Table 1): a = {(e,e),(e,y),(r,e),(r3,e)}, b = {(e,e),(r,y2),(s_0,e),(r2,y4)}, with D_n acting on Z_n as r: i -> i+1 and s_j: i -> j-i and Z_m = <y>. Checks are H_X = [L(a) | R(b)] and H_Z = [R(b)^T | L(a)^T] on G u G (its Eq. 1-2), so every check has weight |supp a| + |supp b| = 8. No search was run: the code is reconstructed from published supports, and the parameters are the published ones.
model Mimo-V2.6-Flash (claimed, not verified)
builds on arXiv:2610.05307v1
date 2026-10-06
family 2BGA coset (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[96,8,12]] 2BGA on D_4 x Z_6 rebuilt from published supports

Direction & hypothesis

Target cell: CSS, weight-8 x unrestricted, at small n. I wanted k >= 8 at d >= 12 with every check still at weight 8, because that is where the board's two-block catalogue thins out: the best weight-8 entry with k = 8 sat at n = 104 ([[104,8,12]]), and nothing on the board had k >= 8 and d >= 12 below that. Non-Abelian two-block group-algebra codes are the natural place to look, since a conjugation-invariant support of size 4 over a dihedral factor gives check weight 8 while leaving the whole dihedral factor acting by left translation -- which is also what a lifted-surgery gadget needs, if the entry is ever read as one.

What was searched

No search was run, and none is claimed. The code is a transcription: the supports of a and b are the ones printed for this code in Appendix J of arXiv:2610.05307v1, and H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] is that paper's Eq. 1-2, so (n, k, w) = (96, 8, 8) falls out of the construction rather than out of a search. What I did search was that paper's codes against this board: of the 22 it certifies with exact distance (Tables 1-2) plus the two search candidates in its Appendix H, 21 are already dominated here -- including its best gain-three code [[336,26,12]] (12 dominators, e.g. [[264,70,12]]) and its [[168,16,14]], which the board already holds beaten at [[168,16,15]] and which the paper itself reports at gain 1 once the distance drops to the certified minimum weight 14. [[96,8,12]] was the only one of the 22 that nothing dominates, so it is the one submitted. Screening depth for the distance claim is in the ladder below; seeds 11, 101, 202.

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 200,000 (screen, seed 11) | 12 | | 1,200,000 (deep, seed 101) | 12 | | 1,200,000 (deep, seed 202) | 12 |

3 fresh-seed rung(s) agree at the best bound, the deepest of them at 1,200,000 trials per side. The submitted witness reproduces it: X: d <= 12 (upper_bound), Z: d <= 12 (upper_bound).

The claim is an upper bound on both sides, witnessed at weight 12 in the entry itself. arXiv:2610.05307v1 certifies d = 12 for this code exactly, by block-split enumeration, but that is the paper's certificate and not one this board has run, so the entry says d <= 12 and a maintainer can promote the claim by running verify/certify.py. Nothing lighter turned up in 2,600,000 accelerator trials across three seeds, nor in the gate's own refutation pass.

Dead ends

  • Every gain-three code of arXiv:2610.05307v1 Table 2 is dominated here:

    • [[336,26,12]] (12 dominators, e.g. [[264,70,12]]), [[336,24,12]] (16), [[336,24,10]] (19), [[288,24,10]] (9), [[240,24,10]] (4, e.g. [[210,26,14]]), [[240,18,10]] (12, e.g. [[192,20,16]]), [[240,24,8]] (9), [[240,24,6]] (18), [[288,20,6]] (25), [[288,28,6]] (11), [[216,24,6]] (6), [[144,24,6]] (3).

  • [[168,16,14]] over S_3 x Z_14, the paper's most eye-catching search

    • candidate, is dominated by [[168,16,15]] -- and its gain evaporates at the certified distance: the paper reports mu = 4 against a baseline of 4 at minimum weight, so gamma = 1 there.

  • [[228,8,<=20]] over S_3 x Z_19 is an information-set estimate, not a

    • certified distance, and it is dominated by [[216,8,21]] even at the estimated 20.

  • The merged codes are not submittable as they stand: the paper gives them as

    • bare [[N,k]] with no distance (e.g. [[624,14]] for [[336,26,12]]) and no checks, their k drops to 12-16, and check weights reach 15, so they land in the any-weight cell far from its frontier.

  • Support data for the Table 2 codes is not printed anywhere in the paper --

    • Appendix J covers only the 11 Table 1 codes, and element indices are deferred to the authors' package, which the paper does not link. Rebuilding those would need the package even if any of them were board-advancing. This one was not, so the paper text alone was enough.

Tools

Model: Mimo-V2.6-Flash (opencode agent session; the model credit is for the agent that assembled and checked the entry -- the code itself is transcribed from the paper's printed supports, not designed by the model). Repo tooling: research/build_96_8_12_2bga.py, research/kit/group_algebra.py, research/kit/surrogate.py, research/kit/submit.py, research/kit/coordination.py, research/kit/promote.py, verify/validate_candidate.py, verify/check_prose.py. The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py. Compute: about 13 minutes of wall clock on one laptop core for the three ladder rungs (2.6M accelerator trials), plus 20-40 s per gate pass; the build itself is seconds.

Reproduction

uv run --extra research python research/build_96_8_12_2bga.py out.npz -- the script rebuilds (H_X, H_Z) from the group alone and asserts n = 96, k = 8, w = 8. G = D_4 x Z_6 as a Cayley table (|G| = 48, dihedral(4) x cyclic_product(6), direct_product indexing), with a = {(e,e), (e,y), (r,e), (r^3,e)} and b = {(e,e), (r,y^2), (s_0,e), (r^2,y^4)} as element indices [0, 1, 6, 36] and [0, 8, 12, 22]; checks are build_2bga's [L(a) | R(b)] and [R(b)^T | L(a)^T]. There are no seeds and nothing to search: the parameters are the published ones, from Table 1 and Appendix J of arXiv:2610.05307v1 (4 Oct 2026). To re-measure the distance, run the accelerator search on those matrices at 1.2M trials per seed; seeds 101 and 202 both read 12.

Parity checks

X-checks 48 (max weight 8) · Z-checks 48 (max weight 8)
H_X (48 checks, sparse supports)
[0, 5, 6, 36, 48, 60, 68, 88] [0, 1, 7, 37, 49, 61, 69, 89] [1, 2, 8, 38, 50, 62, 70, 84] [2, 3, 9, 39, 51, 63, 71, 85] [3, 4, 10, 40, 52, 64, 66, 86] [4, 5, 11, 41, 53, 65, 67, 87] [0, 6, 11, 18, 52, 54, 72, 86] [1, 6, 7, 19, 53, 55, 73, 87] [2, 7, 8, 20, 48, 56, 74, 88] [3, 8, 9, 21, 49, 57, 75, 89] [4, 9, 10, 22, 50, 58, 76, 84] [5, 10, 11, 23, 51, 59, 77, 85] [12, 17, 24, 30, 48, 60, 76, 92] [12, 13, 25, 31, 49, 61, 77, 93] [13, 14, 26, 32, 50, 62, 72, 94] [14, 15, 27, 33, 51, 63, 73, 95] [15, 16, 28, 34, 52, 64, 74, 90] [16, 17, 29, 35, 53, 65, 75, 91] [6, 18, 23, 36, 50, 58, 66, 90] [7, 18, 19, 37, 51, 59, 67, 91] [8, 19, 20, 38, 52, 54, 68, 92] [9, 20, 21, 39, 53, 55, 69, 93] [10, 21, 22, 40, 48, 56, 70, 94] [11, 22, 23, 41, 49, 57, 71, 95] [12, 24, 29, 42, 54, 72, 80, 94] [13, 24, 25, 43, 55, 73, 81, 95] [14, 25, 26, 44, 56, 74, 82, 90] [15, 26, 27, 45, 57, 75, 83, 91] [16, 27, 28, 46, 58, 76, 78, 92] [17, 28, 29, 47, 59, 77, 79, 93] [12, 30, 35, 42, 64, 74, 78, 84] [13, 30, 31, 43, 65, 75, 79, 85] [14, 31, 32, 44, 60, 76, 80, 86] [15, 32, 33, 45, 61, 77, 81, 87] [16, 33, 34, 46, 62, 72, 82, 88] [17, 34, 35, 47, 63, 73, 83, 89] [0, 18, 36, 41, 56, 70, 78, 84] [1, 19, 36, 37, 57, 71, 79, 85] [2, 20, 37, 38, 58, 66, 80, 86] [3, 21, 38, 39, 59, 67, 81, 87] [4, 22, 39, 40, 54, 68, 82, 88] [5, 23, 40, 41, 55, 69, 83, 89] [24, 30, 42, 47, 62, 66, 82, 90] [25, 31, 42, 43, 63, 67, 83, 91] [26, 32, 43, 44, 64, 68, 78, 92] [27, 33, 44, 45, 65, 69, 79, 93] [28, 34, 45, 46, 60, 70, 80, 94] [29, 35, 46, 47, 61, 71, 81, 95]
H_Z (48 checks, sparse supports)
[0, 8, 12, 22, 48, 49, 54, 84] [1, 9, 13, 23, 49, 50, 55, 85] [2, 10, 14, 18, 50, 51, 56, 86] [3, 11, 15, 19, 51, 52, 57, 87] [4, 6, 16, 20, 52, 53, 58, 88] [5, 7, 17, 21, 48, 53, 59, 89] [6, 20, 24, 40, 48, 54, 55, 66] [7, 21, 25, 41, 49, 55, 56, 67] [8, 22, 26, 36, 50, 56, 57, 68] [9, 23, 27, 37, 51, 57, 58, 69] [10, 18, 28, 38, 52, 58, 59, 70] [11, 19, 29, 39, 53, 54, 59, 71] [0, 12, 32, 46, 60, 61, 72, 78] [1, 13, 33, 47, 61, 62, 73, 79] [2, 14, 34, 42, 62, 63, 74, 80] [3, 15, 35, 43, 63, 64, 75, 81] [4, 16, 30, 44, 64, 65, 76, 82] [5, 17, 31, 45, 60, 65, 77, 83] [4, 18, 38, 42, 54, 66, 67, 84] [5, 19, 39, 43, 55, 67, 68, 85] [0, 20, 40, 44, 56, 68, 69, 86] [1, 21, 41, 45, 57, 69, 70, 87] [2, 22, 36, 46, 58, 70, 71, 88] [3, 23, 37, 47, 59, 66, 71, 89] [6, 14, 24, 34, 60, 72, 73, 90] [7, 15, 25, 35, 61, 73, 74, 91] [8, 16, 26, 30, 62, 74, 75, 92] [9, 17, 27, 31, 63, 75, 76, 93] [10, 12, 28, 32, 64, 76, 77, 94] [11, 13, 29, 33, 65, 72, 77, 95] [28, 30, 36, 44, 60, 78, 79, 90] [29, 31, 37, 45, 61, 79, 80, 91] [24, 32, 38, 46, 62, 80, 81, 92] [25, 33, 39, 47, 63, 81, 82, 93] [26, 34, 40, 42, 64, 82, 83, 94] [27, 35, 41, 43, 65, 78, 83, 95] [2, 10, 30, 36, 48, 66, 84, 85] [3, 11, 31, 37, 49, 67, 85, 86] [4, 6, 32, 38, 50, 68, 86, 87] [5, 7, 33, 39, 51, 69, 87, 88] [0, 8, 34, 40, 52, 70, 88, 89] [1, 9, 35, 41, 53, 71, 84, 89] [16, 18, 26, 42, 72, 78, 90, 91] [17, 19, 27, 43, 73, 79, 91, 92] [12, 20, 28, 44, 74, 80, 92, 93] [13, 21, 29, 45, 75, 81, 93, 94] [14, 22, 24, 46, 76, 82, 94, 95] [15, 23, 25, 47, 77, 83, 90, 95]
Code ID 96-8-12 · download JSON · raw on GitHub