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[[144,12,12]] d ≤
n
144
k
12
d
12
kd²/n
12.0
w
6
X/Z
1
g
0.0059
r
6.7082
layers
2
swaps
1051

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · w_X = 6, w_Z = 6 (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 operator (support, 12 qubits)
[11, 21, 23, 50, 52, 62, 65, 72, 76, 77, 103, 134]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[19, 48, 52, 53, 60, 72, 76, 102, 104, 117, 134, 137]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×144 (2,4)×1080 (3,3)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 144 (2,4): 1080 (3,3): 144 (3,5): 10368 (3,7): 1440
trapping sets H_Z (1,3)×144 (2,4)×1080 (3,3)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 144 (2,4): 1080 (3,3): 144 (3,5): 10368 (3,7): 1440
witness diameter X 7.2111 · Z 9.0554 (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
r = 6.708
X checkZ checkqubit site (82)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 1051 nearest-neighbor SWAPs per round in total, at most 14 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 @fullymiddleaged
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction The gross code of Bravyi, Cross, Gambetta, Maslov, Rall and Yoder (arXiv:2308.07915), written on Z_12 x Z_6 as A = y + y2 + x3, B = y3 + x2 + x7 (a relabelling of their A = x3 + y + y2, B = y3 + x + x2). New here: a bilayer layout (max check diameter 6.708) found by simulated annealing from a folded-torus start.
model Claude Claude Opus 5.5 (claimed, not verified)
builds on arXiv:2308.07915
date 2026-09-30
notes Equivalent to the board's 144-12-12.json (IBM's gross code, arXiv:2308.07915): same spectral fingerprint and WL signature. The code is theirs; this entry contributes only the bilayer layout within the 7.0 cap, which the board did not have. Parameters are known (novelty: known_parameters).
family bivariate bicycle (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

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

A bilayer layout of the gross code [[144,12,12]] (Bravyi et al., arXiv:2308.07915)

Direction & hypothesis

The code is not ours. It is IBM's gross code, already on the board as the baseline codes/144-12-12.json, which has no layout and so competes only in the unrestricted cells. The contribution here is a layout: two layers, at most two qubits per site, largest check diameter 6.708, within the 7.0 bilayer cap. With it the gross code enters the 2D-local bilayer weight-6 cell, where no [[144,12,12]] was listed.

What was searched

  • Our screen for the bilayer weight-6 cell (random weight-6 BB genomes on thin tori, 12x6 to 30x6) drew the genome

    • A = y + y^2 + x^3, B = y^3 + x^2 + x^7 on Z_12 x Z_6. It is the gross code relabelled: its spectral fingerprint equals that of the board's 144-12-12 and of IBM's form A = x^3 + y + y^2, B = y^3 + x + x^2, and the submission validator labels it possibly equivalent to 144-12-12.json by WL signature. We did not find an explicit qubit map.

  • The layout came from simulated annealing from a folded-torus start (two qubits per site, minimising the largest

    • check diameter). In IBM's own form the folded-torus start has diameter 7.81, above the cap; this relabelling annealed to 6.708.

Evidence trail

  • The logical witnesses in the submission JSON show d <= 12 on both sides; the board's 144-12-12 is certified d = 12.
  • Our own MILP (scipy / HiGHS) proved no nontrivial Z-type logical of weight below 12 for this genome in 64 s, with

    • d_X = d_Z from a verified X/Z duality permutation. DistQLDPC (arXiv:2606.12445) gave 12 as an independent check.

  • Layout: coordinates in the submission JSON under locality.coordinates; the verifier measures the interaction

    • radius and site spacing from them.

Dead ends

  • With this entry our [[144,8,12]] bilayer entry (PR #2337) is dominated. Its note said we had not laid the gross code

    • out within 7.0; that was true when written and is now superseded by this layout.

  • Laying out other groups' codes is not our main line: of our submissions this is the only one whose code is not

    • ours, and we submit it only because the layout itself is new to the board.

Tools

Claude Opus 5.5 in Claude Code, with our numpy/scipy research package (BB construction, screen, annealing layout, MILP), DistQLDPC (github.com/guluchen/DistQLDPC), and this repository's cli/qldpc.py and verify/.

Reproduction

Take l = 12 and m = 6, with A = y + y^2 + x^3 and B = y^3 + x^2 + x^7 in F_2[x,y]/(x^12 - 1, y^6 - 1). Then H_X = [A | B] and H_Z = [B^T | A^T]. The bilayer coordinates are in the submission JSON.

Parity checks

X-checks 72 (max weight 6) · Z-checks 72 (max weight 6)
H_X (72 checks, sparse supports)
[1, 2, 18, 75, 84, 114] [2, 3, 19, 76, 85, 115] [3, 4, 20, 77, 86, 116] [4, 5, 21, 72, 87, 117] [0, 5, 22, 73, 88, 118] [0, 1, 23, 74, 89, 119] [7, 8, 24, 81, 90, 120] [8, 9, 25, 82, 91, 121] [9, 10, 26, 83, 92, 122] [10, 11, 27, 78, 93, 123] [6, 11, 28, 79, 94, 124] [6, 7, 29, 80, 95, 125] [13, 14, 30, 87, 96, 126] [14, 15, 31, 88, 97, 127] [15, 16, 32, 89, 98, 128] [16, 17, 33, 84, 99, 129] [12, 17, 34, 85, 100, 130] [12, 13, 35, 86, 101, 131] [19, 20, 36, 93, 102, 132] [20, 21, 37, 94, 103, 133] [21, 22, 38, 95, 104, 134] [22, 23, 39, 90, 105, 135] [18, 23, 40, 91, 106, 136] [18, 19, 41, 92, 107, 137] [25, 26, 42, 99, 108, 138] [26, 27, 43, 100, 109, 139] [27, 28, 44, 101, 110, 140] [28, 29, 45, 96, 111, 141] [24, 29, 46, 97, 112, 142] [24, 25, 47, 98, 113, 143] [31, 32, 48, 72, 105, 114] [32, 33, 49, 73, 106, 115] [33, 34, 50, 74, 107, 116] [34, 35, 51, 75, 102, 117] [30, 35, 52, 76, 103, 118] [30, 31, 53, 77, 104, 119] [37, 38, 54, 78, 111, 120] [38, 39, 55, 79, 112, 121] [39, 40, 56, 80, 113, 122] [40, 41, 57, 81, 108, 123] [36, 41, 58, 82, 109, 124] [36, 37, 59, 83, 110, 125] [43, 44, 60, 84, 117, 126] [44, 45, 61, 85, 118, 127] [45, 46, 62, 86, 119, 128] [46, 47, 63, 87, 114, 129] [42, 47, 64, 88, 115, 130] [42, 43, 65, 89, 116, 131] [49, 50, 66, 90, 123, 132] [50, 51, 67, 91, 124, 133] [51, 52, 68, 92, 125, 134] [52, 53, 69, 93, 120, 135] [48, 53, 70, 94, 121, 136] [48, 49, 71, 95, 122, 137] [0, 55, 56, 96, 129, 138] [1, 56, 57, 97, 130, 139] [2, 57, 58, 98, 131, 140] [3, 58, 59, 99, 126, 141] [4, 54, 59, 100, 127, 142] [5, 54, 55, 101, 128, 143] [6, 61, 62, 72, 102, 135] [7, 62, 63, 73, 103, 136] [8, 63, 64, 74, 104, 137] [9, 64, 65, 75, 105, 132] [10, 60, 65, 76, 106, 133] [11, 60, 61, 77, 107, 134] [12, 67, 68, 78, 108, 141] [13, 68, 69, 79, 109, 142] [14, 69, 70, 80, 110, 143] [15, 70, 71, 81, 111, 138] [16, 66, 71, 82, 112, 139] [17, 66, 67, 83, 113, 140]
H_Z (72 checks, sparse supports)
[3, 30, 60, 76, 77, 126] [4, 31, 61, 72, 77, 127] [5, 32, 62, 72, 73, 128] [0, 33, 63, 73, 74, 129] [1, 34, 64, 74, 75, 130] [2, 35, 65, 75, 76, 131] [9, 36, 66, 82, 83, 132] [10, 37, 67, 78, 83, 133] [11, 38, 68, 78, 79, 134] [6, 39, 69, 79, 80, 135] [7, 40, 70, 80, 81, 136] [8, 41, 71, 81, 82, 137] [0, 15, 42, 88, 89, 138] [1, 16, 43, 84, 89, 139] [2, 17, 44, 84, 85, 140] [3, 12, 45, 85, 86, 141] [4, 13, 46, 86, 87, 142] [5, 14, 47, 87, 88, 143] [6, 21, 48, 72, 94, 95] [7, 22, 49, 73, 90, 95] [8, 23, 50, 74, 90, 91] [9, 18, 51, 75, 91, 92] [10, 19, 52, 76, 92, 93] [11, 20, 53, 77, 93, 94] [12, 27, 54, 78, 100, 101] [13, 28, 55, 79, 96, 101] [14, 29, 56, 80, 96, 97] [15, 24, 57, 81, 97, 98] [16, 25, 58, 82, 98, 99] [17, 26, 59, 83, 99, 100] [18, 33, 60, 84, 106, 107] [19, 34, 61, 85, 102, 107] [20, 35, 62, 86, 102, 103] [21, 30, 63, 87, 103, 104] [22, 31, 64, 88, 104, 105] [23, 32, 65, 89, 105, 106] [24, 39, 66, 90, 112, 113] [25, 40, 67, 91, 108, 113] [26, 41, 68, 92, 108, 109] [27, 36, 69, 93, 109, 110] [28, 37, 70, 94, 110, 111] [29, 38, 71, 95, 111, 112] [0, 30, 45, 96, 118, 119] [1, 31, 46, 97, 114, 119] [2, 32, 47, 98, 114, 115] [3, 33, 42, 99, 115, 116] [4, 34, 43, 100, 116, 117] [5, 35, 44, 101, 117, 118] [6, 36, 51, 102, 124, 125] [7, 37, 52, 103, 120, 125] [8, 38, 53, 104, 120, 121] [9, 39, 48, 105, 121, 122] [10, 40, 49, 106, 122, 123] [11, 41, 50, 107, 123, 124] [12, 42, 57, 108, 130, 131] [13, 43, 58, 109, 126, 131] [14, 44, 59, 110, 126, 127] [15, 45, 54, 111, 127, 128] [16, 46, 55, 112, 128, 129] [17, 47, 56, 113, 129, 130] [18, 48, 63, 114, 136, 137] [19, 49, 64, 115, 132, 137] [20, 50, 65, 116, 132, 133] [21, 51, 60, 117, 133, 134] [22, 52, 61, 118, 134, 135] [23, 53, 62, 119, 135, 136] [24, 54, 69, 120, 142, 143] [25, 55, 70, 121, 138, 143] [26, 56, 71, 122, 138, 139] [27, 57, 66, 123, 139, 140] [28, 58, 67, 124, 140, 141] [29, 59, 68, 125, 141, 142]
Code ID 144-12-12-b · download JSON · raw on GitHub