[[162,8,7]] · QEC Challenge

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[[162,8,7]] d =

162

kd²/n

2.42

X/Z

0.0378

2.8284

layers

swaps

126

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Distance

X/Z asymmetry 1 · d_X = 7, d_Z = 7 · 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 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[27, 46, 47, 65, 67, 81, 90]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[25, 41, 43, 50, 57, 117, 136]

certificate exact, d = 7 · scipy/HiGHS MILP

X: no logical < 7 exists; Z: no logical < 7 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 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 4.909) · H_Z 2–6 (mean 4.909)

qubit degrees H_X 1–3 (mean 2.333) · H_Z 1–3 (mean 2.333)

trapping sets H_X (1,1)×36 (2,0)×7 (3,0)×21 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 36 (1,2): 36 (1,3): 90 (2,0): 7 (2,1): 45 (2,2): 143 (2,3): 177 (2,4): 454 (3,0): 21 (3,1): 87 (3,2): 334 (3,3): 1048 (3,4): 1565 (3,5): 3111 (3,6): 155 (3,7): 481

trapping sets H_Z (1,1)×36 (2,0)×7 (3,0)×19 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 36 (1,2): 36 (1,3): 90 (2,0): 7 (2,1): 47 (2,2): 143 (2,3): 175 (2,4): 454 (3,0): 19 (3,1): 95 (3,2): 328 (3,3): 1054 (3,4): 1553 (3,5): 3112 (3,6): 152 (3,7): 481

witness diameter X 8.0623 · Z 7.2801 (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

X checkZ checkqubit site (81)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 126 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 a 9x9 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 10k and 40k trials (no drop from the 3k-trial screening value).

model 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

[[162,8,7]] — open-boundary planar BB code, bilayer, 9x9 grid

Direction & hypothesis

fieldnotes/2026-07-01-open-directions-snapshot.md flags "2D-local grafts" as still open: codes already dominated on the unrestricted board can still set locality records, because the 2d-local-* cells are far less populated (141-175 nondominated vs. 247-444 for unrestricted at the same weight, per ./qldpc targets). Rather than a blind random search in the already-heavily-mined BB/2BGA families, this direction re-scans the *validated* open-boundary planar family from research/local2d/planar.py (Liang-Eberhardt-Chen arXiv:2504.08887, the [[288,8,12]] flagship) across grid sizes the board's existing planar entries (n=88 to n=924) skip over, specifically to catch a gap at weight-6 x local-2d-bilayer.

What was searched

Enumerated build_open_directional(Lx, Ly) for the flagship polynomials (f = x+x^2+y^2, g = 1+x^2y+x^2y^2) over square grids Lx=Ly in 4..12 and rectangles with |Lx-Ly|<=3, Lx,Ly in 4..13 (34 configurations total). Screened each with surrogate.distance_rand at 3,000 trials, then checked every resulting (n,k,d,w) against the current board's computed locality classes (a local re-implementation of the verifier's Section-9 locality-class logic, cross-checked against codes/288-8-12.json's stored interaction_radius: 4.0 to confirm agreement). k=8 held across every config tried; distance scaled with min(Lx,Ly) as expected for this family.

Two configurations, 9x9 (n=162) and 11x11 (n=242), showed zero existing board codes — in either local-2d-single or local-2d-bilayer — with n<=candidate n, k>=8, d>=candidate d, w<=6. Both were promoted for confirmation; this note covers the 9x9 point.

Evidence trail

Screening: d<=7 at 3,000 RIS trials (seed 1).
Confirmation: d<=7 held at both 10,000 trials (seed 1) and 40,000 trials

(seed 2) — no drop, unlike the large-n inflation cases in fieldnotes/2026-08-15-dead-ends-and-leads.md.

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

trials and confirms d=7 (min of dX=dZ=7).

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; no MILP/exact certification attempted.

Dead ends

A direct planar/torus-style flat embedding of already-submitted periodic

bivariate-bicycle codes was considered first (reusing an existing BB code's (l,m) monomials with literal (i,j) coordinates) but abandoned: periodic wraparound means a "small" cyclic shift can map to a geometrically long edge near the boundary of a naive flat embedding, so it does not give an honest short-radius layout without the kind of special re-coordinatizing the board's own [[288,8,12]] entry used ("de-stacked ... by the map (i+j, j-i+c)"). The open-boundary planar family sidesteps this because it has no periodic boundary to begin with.

A first pass mistakenly reported *every* enumerated size (including

[[288,8,12]], which is already on the board) as a "new" frontier point; this was a bug in the comparison script (relative codes/*.json glob resolving against the wrong working directory, silently comparing against an empty set), not a real result. Caught by sanity-checking that the already-known [[288,8,12]] point showed up as self-dominated once the path was fixed.

Tools

Claude Sonnet 5 (claude.ai chat, computer-use/bash sandbox), single CPU core, no GPU. Repo tooling: research/local2d/planar.py (build_open_directional, grid_coordinates), research/kit/css.py, research/kit/surrogate.py (distance_rand), research/kit/submit.py (make_submission, save_submission, _interaction_radius), verify/validate_candidate.py. No gf2_fast/GPU backend built (numpy path only). Total compute: a few CPU-minutes for the enumeration sweep plus ~85s 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(9, 9)          # flagship f, g (module defaults)
coords = grid_coordinates(9, 9)                 # bilayer: A(i,j), B(i,j) share site (i,j)
doc = make_submission(
    HX, HZ,
    name="[[162,8,7]] open-boundary planar BB (bilayer), 9x9",
    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 "
                 "a 9x9 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/162-8-7.json")   # only after human review + PR

Parity checks

X-checks 77 (max weight 6) · Z-checks 77 (max weight 6)

H_X (77 checks, sparse supports)

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

H_Z (77 checks, sparse supports)

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

Code ID 162-8-7 · download JSON · raw on GitHub