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[[242,8,10]] d ≤
n
242
k
8
d
10
kd²/n
3.306
w
6
X/Z
1
g
0.0517
r
2.8284
layers
2
swaps
198

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Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · 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 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[55, 78, 79, 101, 103, 121, 132, 134, 143, 154]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[42, 62, 64, 73, 82, 132, 155, 177, 178, 201]
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 5.077) · H_Z 2–6 (mean 5.077)
qubit degrees H_X 1–3 (mean 2.455) · H_Z 1–3 (mean 2.455)
trapping sets H_X (1,1)×44 (2,0)×9 (3,0)×27 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 44 (1,2): 44 (1,3): 154 (2,0): 9 (2,1): 57 (2,2): 179 (2,3): 221 (2,4): 866 (3,0): 27 (3,1): 113 (3,2): 428 (3,3): 1380 (3,4): 2007 (3,5): 6587 (3,6): 197 (3,7): 985
trapping sets H_Z (1,1)×44 (2,0)×9 (3,0)×25 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 44 (1,2): 44 (1,3): 154 (2,0): 9 (2,1): 59 (2,2): 179 (2,3): 219 (2,4): 866 (3,0): 25 (3,1): 121 (3,2): 422 (3,3): 1386 (3,4): 1995 (3,5): 6588 (3,6): 194 (3,7): 985
witness diameter X 9.8489 · Z 9.8489 (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 = 2.828
X checkZ checkqubit site (121)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 198 nearest-neighbor SWAPs per round in total, at most 1 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 @msilve160
provenance submitted through the challenge
novelty novelty not audited
construction Open-boundary directional-condensation planar bivariate-bicycle code (Liang-Eberhardt-Chen arXiv:2504.08887, flagship f=x+x2+y2, g=1+x2y+x2y2) on an 11x11 grid; built with research/local2d/planar.build_open_directional (validated construction, greedy footnote-6 corner resolution). Distance is a Monte-Carlo upper bound from research/kit/surrogate.distance_rand, confirmed stable at 15k trials (no drop from the 3k-trial screening value).
model Claude Claude Sonnet 5 (claimed, not verified)
builds on arXiv:2504.08887
date 2026-09-18
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

[[242,8,10]] — open-boundary planar BB code, bilayer, 11x11 grid

Direction & hypothesis

Same direction as 162-8-7.note.md: the still-open "2D-local grafts" lead from fieldnotes/2026-07-01-open-directions-snapshot.md, re-scanning the validated open-boundary planar family (research/local2d/planar.py, Liang-Eberhardt-Chen arXiv:2504.08887 flagship) at grid sizes the board's existing planar entries (n=88..924) don't cover, targeting the thinner weight-6 x local-2d-bilayer cell rather than the heavily-mined periodic BB/2BGA families.

What was searched

Same 34-configuration sweep described in 162-8-7.note.md (square grids Lx=Ly in 4..12 and rectangles with |Lx-Ly|<=3 in 4..13). The 11x11 (n=242) configuration was the second of two points (alongside 9x9/n=162) with zero existing board codes dominating it in either local-2d-single or local-2d-bilayer at weight<=8.

Evidence trail

  • Screening: d<=10 at 3,000 RIS trials (seed 1).
  • Confirmation: d<=10 held at 15,000 trials (seed 3) — no drop from the

    • screening value.

  • Packaging: submit.make_submission re-searched the witness at 20,000

    • trials and confirms d=10 (min of dX=dZ=10).

  • Gate: verify/validate_candidate.py on the packaged doc returned

    • passed: true, refute.refuted: false (no lighter logical in a further 8,000 independent RIS trials), and novelty.board_advancing: true for cell (weight-6, local-2d-bilayer) with dominated_by: [].

  • Claim: witness-backed upper bound, not exact.

    • confidence: "upper_bound" in the packaged doc.

Dead ends

See 162-8-7.note.md — same sweep, same script-path bug caught and fixed before either candidate was trusted.

One additional note specific to this size: confirmation at deeper trial counts (e.g. the ~40k used for the 9x9 point) was not run here due to the single-core sandbox's time budget — n=242 costs roughly 4x the per-trial time of n=162, so only a 15k-trial re-check was affordable in this session. A human continuing this should push to at least 40k-60k trials/side before treating d=10 as well-supported, per the trial-depth floor in fieldnotes/2026-07-01-trial-depth-floors.md.

Tools

Claude Sonnet 5 (claude.ai chat, computer-use/bash sandbox), single CPU core, no GPU. Same tool chain as 162-8-7.note.md. Total compute: shared enumeration sweep plus ~165s for final packaging of this candidate.

Reproduction

import sys
sys.path.insert(0, "research/local2d")
sys.path.insert(0, "research/kit")
from planar import build_open_directional, grid_coordinates
from submit import make_submission, save_submission

HX, HZ = build_open_directional(11, 11)
coords = grid_coordinates(11, 11)
doc = make_submission(
    HX, HZ,
    name="[[242,8,10]] open-boundary planar BB (bilayer), 11x11",
    construction="Open-boundary directional-condensation planar bivariate-bicycle "
                 "code (arXiv:2504.08887 flagship f=x+x^2+y^2, g=1+x^2y+x^2y^2) on "
                 "an 11x11 grid; built with research/local2d/planar.build_open_directional.",
    authors=["@msilve160"], family="bivariate-bicycle",
    references=["arXiv:2504.08887"], confidence="upper_bound",
    coordinates=coords, layers=2, trials=20000, seed=0,
)
save_submission(doc, "codes/242-8-10.json")   # only after human review + PR

Parity checks

X-checks 117 (max weight 6) · Z-checks 117 (max weight 6)
H_X (117 checks, sparse supports)
[0, 122, 123] [1, 123, 124] [2, 124, 125] [3, 125, 126] [4, 126, 127] [5, 127, 128] [6, 128, 129] [7, 129, 130] [8, 130, 131] [0, 11, 133, 134] [1, 12, 134, 135] [2, 13, 135, 136] [3, 14, 136, 137] [4, 15, 137, 138] [5, 16, 138, 139] [6, 17, 139, 140] [7, 18, 140, 141] [8, 19, 141, 142] [2, 11, 22, 121, 144, 145] [3, 12, 23, 122, 145, 146] [4, 13, 24, 123, 146, 147] [5, 14, 25, 124, 147, 148] [6, 15, 26, 125, 148, 149] [7, 16, 27, 126, 149, 150] [8, 17, 28, 127, 150, 151] [9, 18, 29, 128, 151, 152] [10, 19, 30, 129, 152, 153] [13, 22, 33, 132, 155, 156] [14, 23, 34, 133, 156, 157] [15, 24, 35, 134, 157, 158] [16, 25, 36, 135, 158, 159] [17, 26, 37, 136, 159, 160] [18, 27, 38, 137, 160, 161] [19, 28, 39, 138, 161, 162] [20, 29, 40, 139, 162, 163] [21, 30, 41, 140, 163, 164] [24, 33, 44, 143, 166, 167] [25, 34, 45, 144, 167, 168] [26, 35, 46, 145, 168, 169] [27, 36, 47, 146, 169, 170] [28, 37, 48, 147, 170, 171] [29, 38, 49, 148, 171, 172] [30, 39, 50, 149, 172, 173] [31, 40, 51, 150, 173, 174] [32, 41, 52, 151, 174, 175] [35, 44, 55, 154, 177, 178] [36, 45, 56, 155, 178, 179] [37, 46, 57, 156, 179, 180] [38, 47, 58, 157, 180, 181] [39, 48, 59, 158, 181, 182] [40, 49, 60, 159, 182, 183] [41, 50, 61, 160, 183, 184] [42, 51, 62, 161, 184, 185] [43, 52, 63, 162, 185, 186] [46, 55, 66, 165, 188, 189] [47, 56, 67, 166, 189, 190] [48, 57, 68, 167, 190, 191] [49, 58, 69, 168, 191, 192] [50, 59, 70, 169, 192, 193] [51, 60, 71, 170, 193, 194] [52, 61, 72, 171, 194, 195] [53, 62, 73, 172, 195, 196] [54, 63, 74, 173, 196, 197] [57, 66, 77, 176, 199, 200] [58, 67, 78, 177, 200, 201] [59, 68, 79, 178, 201, 202] [60, 69, 80, 179, 202, 203] [61, 70, 81, 180, 203, 204] [62, 71, 82, 181, 204, 205] [63, 72, 83, 182, 205, 206] [64, 73, 84, 183, 206, 207] [65, 74, 85, 184, 207, 208] [68, 77, 88, 187, 210, 211] [69, 78, 89, 188, 211, 212] [70, 79, 90, 189, 212, 213] [71, 80, 91, 190, 213, 214] [72, 81, 92, 191, 214, 215] [73, 82, 93, 192, 215, 216] [74, 83, 94, 193, 216, 217] [75, 84, 95, 194, 217, 218] [76, 85, 96, 195, 218, 219] [79, 88, 99, 198, 221, 222] [80, 89, 100, 199, 222, 223] [81, 90, 101, 200, 223, 224] [82, 91, 102, 201, 224, 225] [83, 92, 103, 202, 225, 226] [84, 93, 104, 203, 226, 227] [85, 94, 105, 204, 227, 228] [86, 95, 106, 205, 228, 229] [87, 96, 107, 206, 229, 230] [90, 99, 110, 209, 232, 233] [91, 100, 111, 210, 233, 234] [92, 101, 112, 211, 234, 235] [93, 102, 113, 212, 235, 236] [94, 103, 114, 213, 236, 237] [95, 104, 115, 214, 237, 238] [96, 105, 116, 215, 238, 239] [97, 106, 117, 216, 239, 240] [98, 107, 118, 217, 240, 241] [101, 110, 220] [102, 111, 221] [103, 112, 222] [104, 113, 223] [105, 114, 224] [106, 115, 225] [107, 116, 226] [108, 117, 227] [109, 118, 228] [112, 231] [113, 232] [114, 233] [115, 234] [116, 235] [117, 236] [118, 237] [119, 238] [120, 239]
H_Z (117 checks, sparse supports)
[22, 121, 132] [0, 23, 122, 133] [0, 1, 24, 123, 134, 143] [1, 2, 25, 124, 135, 144] [2, 3, 26, 125, 136, 145] [3, 4, 27, 126, 137, 146] [4, 5, 28, 127, 138, 147] [5, 6, 29, 128, 139, 148] [6, 7, 30, 129, 140, 149] [7, 8, 31, 130, 141, 150] [8, 9, 32, 131, 142, 151] [9, 10, 152] [10, 153] [33, 132, 143] [11, 34, 133, 144] [11, 12, 35, 134, 145, 154] [12, 13, 36, 135, 146, 155] [13, 14, 37, 136, 147, 156] [14, 15, 38, 137, 148, 157] [15, 16, 39, 138, 149, 158] [16, 17, 40, 139, 150, 159] [17, 18, 41, 140, 151, 160] [18, 19, 42, 141, 152, 161] [19, 20, 43, 142, 153, 162] [20, 21, 163] [21, 164] [44, 143, 154] [22, 45, 144, 155] [22, 23, 46, 145, 156, 165] [23, 24, 47, 146, 157, 166] [24, 25, 48, 147, 158, 167] [25, 26, 49, 148, 159, 168] [26, 27, 50, 149, 160, 169] [27, 28, 51, 150, 161, 170] [28, 29, 52, 151, 162, 171] [29, 30, 53, 152, 163, 172] [30, 31, 54, 153, 164, 173] [31, 32, 174] [32, 175] [55, 154, 165] [33, 56, 155, 166] [33, 34, 57, 156, 167, 176] [34, 35, 58, 157, 168, 177] [35, 36, 59, 158, 169, 178] [36, 37, 60, 159, 170, 179] [37, 38, 61, 160, 171, 180] [38, 39, 62, 161, 172, 181] [39, 40, 63, 162, 173, 182] [40, 41, 64, 163, 174, 183] [41, 42, 65, 164, 175, 184] [42, 43, 185] [43, 186] [66, 165, 176] [44, 67, 166, 177] [44, 45, 68, 167, 178, 187] [45, 46, 69, 168, 179, 188] [46, 47, 70, 169, 180, 189] [47, 48, 71, 170, 181, 190] [48, 49, 72, 171, 182, 191] [49, 50, 73, 172, 183, 192] [50, 51, 74, 173, 184, 193] [51, 52, 75, 174, 185, 194] [52, 53, 76, 175, 186, 195] [53, 54, 196] [54, 197] [77, 176, 187] [55, 78, 177, 188] [55, 56, 79, 178, 189, 198] [56, 57, 80, 179, 190, 199] [57, 58, 81, 180, 191, 200] [58, 59, 82, 181, 192, 201] [59, 60, 83, 182, 193, 202] [60, 61, 84, 183, 194, 203] [61, 62, 85, 184, 195, 204] [62, 63, 86, 185, 196, 205] [63, 64, 87, 186, 197, 206] [64, 65, 207] [65, 208] [88, 187, 198] [66, 89, 188, 199] [66, 67, 90, 189, 200, 209] [67, 68, 91, 190, 201, 210] [68, 69, 92, 191, 202, 211] [69, 70, 93, 192, 203, 212] [70, 71, 94, 193, 204, 213] [71, 72, 95, 194, 205, 214] [72, 73, 96, 195, 206, 215] [73, 74, 97, 196, 207, 216] [74, 75, 98, 197, 208, 217] [75, 76, 218] [76, 219] [99, 198, 209] [77, 100, 199, 210] [77, 78, 101, 200, 211, 220] [78, 79, 102, 201, 212, 221] [79, 80, 103, 202, 213, 222] [80, 81, 104, 203, 214, 223] [81, 82, 105, 204, 215, 224] [82, 83, 106, 205, 216, 225] [83, 84, 107, 206, 217, 226] [84, 85, 108, 207, 218, 227] [85, 86, 109, 208, 219, 228] [86, 87, 229] [87, 230] [110, 209, 220] [88, 111, 210, 221] [88, 89, 112, 211, 222, 231] [89, 90, 113, 212, 223, 232] [90, 91, 114, 213, 224, 233] [91, 92, 115, 214, 225, 234] [92, 93, 116, 215, 226, 235] [93, 94, 117, 216, 227, 236] [94, 95, 118, 217, 228, 237] [95, 96, 119, 218, 229, 238] [96, 97, 120, 219, 230, 239] [97, 98, 240] [98, 241]
Code ID 242-8-10 · download JSON · raw on GitHub