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[[178,8,9]] d ≤
n
178
k
8
d
9
kd²/n
3.64
w
6
X/Z
1
g
0.0569
r
4.0
layers
1
swaps
1085

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Distance

X/Z asymmetry 1 · d_X ≤ 9, d_Z ≤ 9 · 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 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[45, 66, 67, 87, 94, 96, 104, 113, 173]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[62, 72, 78, 104, 124, 144, 145, 165, 171]
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 2–6 (mean 5.0) · H_Z 2–6 (mean 4.941)
qubit degrees H_X 1–3 (mean 2.388) · H_Z 1–3 (mean 2.36)
trapping sets H_X (1,1)×32 (2,0)×7 (3,0)×18 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 32 (1,2): 45 (1,3): 101 (2,0): 7 (2,1): 55 (2,2): 136 (2,3): 217 (2,4): 520 (3,0): 18 (3,1): 96 (3,2): 392 (3,3): 1024 (3,4): 1860 (3,5): 3660 (3,6): 194 (3,7): 552
trapping sets H_Z (1,1)×34 (2,0)×7 (3,0)×15 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 34 (1,2): 46 (1,3): 98 (2,0): 7 (2,1): 54 (2,2): 142 (2,3): 216 (2,4): 497 (3,0): 15 (3,1): 101 (3,2): 367 (3,3): 1083 (3,4): 1903 (3,5): 3381 (3,6): 203 (3,7): 526
witness diameter X 13.0384 · Z 15.2971 (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 = 4
X checkZ checkqubit site (178)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 1085 nearest-neighbor SWAPs per round in total, at most 8 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 @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Open-boundary planar bivariate-bicycle code (Liang, Eberhardt, Chen, arXiv:2504.08887) with f = x + x2 + y2 and g = 1 + x2 y + x2 y2 on a 10 x 10 grid (research/local2d/planar.py build_open_directional(10, 10), n = 200, k = 8), reduced to n = 178 by 22 restricted r=1 lattice grafts (arXiv:2504.08887 Sec. III E): a qubit lying in exactly one stabilizer of some type is removed together with that stabilizer, accepted only if k stays 8 and a NumPy RIS search screened at 500 fixed-seed trials and confirmed at 1500 fresh-seed trials finds nothing lighter than 9, followed by the weight-1 stabilizer cleanup of research/local2d/boundary_engine.py. Layout: a surviving qubit of unreduced index q = c*100 + i*10 + j sits at (i + j, j - i + c), one layer, spacing 1.
model Claude Claude Fable 5.1 (claimed, not verified)
date 2026-09-20
notes Derived from the flagship planar family at L = (10,10) by r=1 grafting; not an independent construction. Fresh-seed RIS ladder on the saved code: 5000 trials -> 9, 20000 trials -> 9. Claim: d <= 9, a witness-backed upper bound. Advances the weight-6 x local-2d-single board if the gate says so; novelty vs the literature unverified.
family bivariate bicycle (a tag, not a ranking)
locality 2D-local single (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

[[178,8,9]] reduced single-layer planar bivariate-bicycle code, L = (10,10), r=1 grafts only

Direction & hypothesis

Target cell: weight-6 x local-2d-single. Its k = 8 entries are all the flagship open-boundary planar bivariate-bicycle family (f = x + x^2 + y^2, g = 1 + x^2 y + x^2 y^2, arXiv:2504.08887): the unreduced L = 8, 10, 12 lattices and r=1 grafted L = 13 to 16 lattices. The reductions of the smaller lattices on the board (n = 118, 169, 203, 265) use a merge-graft move whose checks exceed radius 4 in the single-layer layout, so they sit in the bilayer cell only. Restricted r=1 grafting removes a qubit together with the one stabilizer of some type that contains it, so every surviving check is a subset of an original one and the single-layer layout (i + j, j - i + c) keeps radius 4 exactly. Hypothesis: the L = 10 lattice grafts to n < 200 at unchanged k = 8 and d = 9. At d = 9 any n < 200 dominates the board's [[200,8,9]] (same k, d and check weight, fewer qubits).

Higher k is closed in this layout: mixed volume over every trinomial (f, g) pair fitting radius 4 under the checkerboard map, a shifted checkerboard, and an anisotropic stripe map never exceeds 8 (3522 to 3562 fitting pairs per map).

What was searched

Base code research/local2d/planar.py build_open_directional(10, 10), [[200,8]]. Grafting driver (staged with the candidate): candidates are qubits lying in exactly one X-stabilizer or one Z-stabilizer; order randomized with seed 1; a removal is kept iff k stays 8, a fixed-seed NumPy RIS screen at 500 trials finds nothing lighter than 9, and a fresh-seed confirm at 1500 trials agrees. 15 removals accepted, 0 rejected at the confirm rung (screen passed, confirm found weight n/a), then the weight-1 stabilizer cleanup of boundary_engine._cleanup. Final n = 178. Wall time 310 s.

Evidence trail

Fresh-seed NumPy RIS ladder on the saved code: 5000 trials -> 9, 20000 trials -> 9. The witnesses in the submission are weight-9 (X) and weight-9 (Z) logicals, re-verified by the GF(2) stack. Claim: d <= 9, a witness-backed upper bound. Gate verdict (verify/validate_candidate.py): passed = True; labels: advances the weight-6 x local-2d-single board; literature novelty UNVERIFIED. Interaction radius 4.0, one layer, one qubit per site, max check weight 6. Advances the weight-6 x local-2d-single board; novelty vs the literature unverified.

Dead ends

Rectangular lattices (10,11) and (11,12): d_rand 9 at n = 220 and 10 at n = 264 (300 trials), dominated by the reductions of the square lattices. Anisotropic single-layer layouts do not open k > 8 (mixed-volume enumeration above). Merge-graft moves were not attempted because they leave the single-layer radius-4 class.

Tools

Claude (Claude Code) as the agent in an unattended autoresearch run. Kit modules research/local2d/planar.py, research/local2d/boundary_engine.py (_cleanup, compute_k), research/kit/surrogate.py (NumPy RIS, no gf2_fast), research/kit/submit.py, and verify/validate_candidate.py as the only gate. Two worker threads on a shared machine.

Reproduction

The graft order is randomized, so the surviving qubit set is recorded by the coordinates in the submission: a qubit at (x, y) has c = (x + y) mod 2, j = (x + y - c) / 2, i = x - j, unreduced index c*100 + i*10 + j. Rebuild the base with

import sys; sys.path.insert(0, "research/local2d")
from planar import build_open_directional
HX, HZ = build_open_directional(10, 10)   # [[200,8]]

then delete the qubits absent from the layout together with the stabilizers that become the unique owner of a deleted qubit, and run boundary_engine._cleanup.

Parity checks

X-checks 85 (max weight 6) · Z-checks 85 (max weight 6)
H_X (85 checks, sparse supports)
[0, 88, 89] [1, 89, 90] [2, 90, 91] [3, 91, 92] [4, 92, 93] [5, 93] [0, 6, 95, 96] [1, 7, 96, 97] [2, 8, 97, 98] [3, 9, 98, 99] [4, 10, 99, 100] [5, 11, 100, 101] [12, 101, 102] [13, 102, 103] [2, 6, 16, 87, 105, 106] [3, 7, 17, 88, 106, 107] [4, 8, 18, 89, 107, 108] [5, 9, 19, 90, 108, 109] [10, 20, 91, 109, 110] [11, 21, 92, 110, 111] [12, 22, 93, 111, 112] [8, 16, 26, 94, 114, 115] [9, 17, 27, 95, 115, 116] [10, 18, 28, 96, 116, 117] [11, 19, 29, 97, 117, 118] [12, 20, 30, 98, 118, 119] [13, 21, 31, 99, 119, 120] [14, 22, 32, 100, 120, 121] [15, 23, 33, 101, 121, 122] [18, 26, 36, 104, 124, 125] [19, 27, 37, 105, 125, 126] [20, 28, 38, 106, 126, 127] [21, 29, 39, 107, 127, 128] [22, 30, 40, 108, 128, 129] [23, 31, 41, 109, 129, 130] [24, 32, 42, 110, 130, 131] [25, 33, 43, 111, 131, 132] [28, 36, 45, 113, 134, 135] [29, 37, 46, 114, 135, 136] [30, 38, 47, 115, 136, 137] [31, 39, 48, 116, 137, 138] [32, 40, 49, 117, 138, 139] [33, 41, 50, 118, 139, 140] [34, 42, 51, 119, 140, 141] [35, 43, 52, 120, 141, 142] [38, 45, 55, 123, 144, 145] [39, 46, 56, 124, 145, 146] [40, 47, 57, 125, 146, 147] [41, 48, 58, 126, 147, 148] [42, 49, 59, 127, 148, 149] [43, 50, 60, 128, 149, 150] [44, 51, 61, 129, 150, 151] [47, 55, 65, 133, 153, 154] [48, 56, 66, 134, 154, 155] [49, 57, 67, 135, 155, 156] [50, 58, 68, 136, 156, 157] [51, 59, 69, 137, 157, 158] [52, 60, 70, 138, 158, 159] [53, 61, 71, 139, 159, 160] [54, 62, 72, 140, 160, 161] [57, 65, 74, 143, 163, 164] [58, 66, 144, 164, 165] [59, 67, 75, 145, 165, 166] [60, 68, 76, 146, 166, 167] [61, 69, 77, 147, 167, 168] [62, 70, 78, 148, 168, 169] [63, 71, 79, 149, 169, 170] [64, 72, 80, 150, 170, 171] [67, 74, 82, 152, 172, 173] [68, 83, 153, 173] [69, 75, 154] [70, 76, 84, 155, 174] [71, 77, 85, 156, 174, 175] [72, 78, 157, 175, 176] [73, 79, 158, 176, 177] [75, 82, 162] [76, 83, 163] [77, 164] [78, 84, 165] [79, 85, 166] [80, 167] [81, 168] [84, 172] [85, 173] [86, 174]
H_Z (85 checks, sparse supports)
[16, 87, 94] [0, 17, 88, 95] [0, 1, 18, 89, 96, 104] [1, 2, 19, 90, 97, 105] [2, 3, 20, 91, 98, 106] [3, 4, 21, 92, 99, 107] [4, 5, 22, 93, 100, 108] [26, 94, 104] [6, 27, 95, 105] [6, 7, 28, 96, 106, 113] [7, 8, 29, 97, 107, 114] [8, 9, 30, 98, 108, 115] [9, 10, 31, 99, 109, 116] [10, 11, 32, 100, 110, 117] [11, 12, 33, 101, 111, 118] [12, 13, 34, 102, 112, 119] [13, 14, 35, 103, 120] [14, 15, 121] [15, 122] [36, 104, 113] [16, 37, 105, 114] [16, 17, 38, 106, 115, 123] [17, 18, 39, 107, 116, 124] [18, 19, 40, 108, 117, 125] [19, 20, 41, 109, 118, 126] [20, 21, 42, 110, 119, 127] [21, 22, 43, 111, 120, 128] [22, 23, 44, 112, 121, 129] [23, 24, 122, 130] [24, 25, 131] [25, 132] [45, 113, 123] [26, 46, 114, 124] [26, 27, 47, 115, 125, 133] [27, 28, 48, 116, 126, 134] [28, 29, 49, 117, 127, 135] [29, 30, 50, 118, 128, 136] [30, 31, 51, 119, 129, 137] [31, 32, 52, 120, 130, 138] [32, 33, 53, 121, 131, 139] [33, 34, 54, 122, 132, 140] [34, 35, 141] [35, 142] [55, 123, 133] [36, 56, 124, 134] [36, 37, 57, 125, 135, 143] [37, 38, 58, 126, 136, 144] [38, 39, 59, 127, 137, 145] [39, 40, 60, 128, 138, 146] [40, 41, 61, 129, 139, 147] [41, 42, 62, 130, 140, 148] [42, 43, 63, 131, 141, 149] [43, 44, 64, 132, 142, 150] [44, 151] [65, 133, 143] [45, 66, 134, 144] [45, 46, 67, 135, 145, 152] [46, 47, 68, 136, 146, 153] [47, 48, 69, 137, 147, 154] [48, 49, 70, 138, 148, 155] [49, 50, 71, 139, 149, 156] [50, 51, 72, 140, 150, 157] [51, 52, 73, 141, 151, 158] [52, 53, 142, 159] [53, 54, 160] [54, 161] [74, 143, 152] [55, 56, 75, 145, 154, 162] [56, 57, 76, 146, 155, 163] [57, 58, 77, 147, 156, 164] [58, 59, 78, 148, 157, 165] [59, 60, 79, 149, 158, 166] [60, 61, 80, 150, 159, 167] [61, 62, 81, 151, 160, 168] [62, 63, 161, 169] [63, 64, 170] [64, 171] [82, 152, 162] [65, 83, 153, 163] [66, 67, 84, 155, 165, 172] [67, 68, 85, 156, 166, 173] [70, 71, 86, 159, 169, 174] [71, 72, 160, 170, 175] [72, 73, 161, 171, 176] [73, 177]
Code ID 178-8-9 · download JSON · raw on GitHub