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[[72,10,9]] d =
n
72
k
10
d
9
kd²/n
11.25
w
8
X/Z
1
g
0.011
r
5.6569
layers
2
swaps
353

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Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · 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 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[8, 12, 18, 31, 47, 52, 57, 59, 71]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[3, 12, 20, 24, 26, 48, 50, 57, 66]
certificate exact, d = 9 · CryptoMiniSat 5.14 SAT
X: no logical < 9 exists; Z: no logical < 9 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)×72 (2,4)×72 (3,4)×108 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 72 (2,4): 72 (2,6): 864 (3,4): 108 (3,6): 3024 (3,8): 12204 (3,10): 1224
trapping sets H_Z (1,4)×72 (2,4)×72 (3,4)×108 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 72 (2,4): 72 (2,6): 864 (3,4): 108 (3,6): 3024 (3,8): 12204 (3,10): 1224
witness diameter X 5.099 · Z 5.6569 (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. · 2026-10-03
r = 5.657
X checkZ checkqubit site (36)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 353 nearest-neighbor SWAPs per round in total, at most 9 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 Published 2BGA database row from arXiv:2306.16400v1: SmallGroup(36,2), a=1+g2+g3+g16, b=1+g6+g17+g21.
builds on arXiv:2306.16400, arXiv:2306.16400 (isomorphic to 2bga:SmallGroup(36,2):a[2,3,16]:b[6,17,21]; research/provenance/derived.json)
date 2026-08-17
notes Published Lin–Pryadko baseline reconstructed from the authors public database at commit 403d194c3f98f0cadc236aecbc4a8b6139ccf23c. Paper reports exact d=9; repository records a witness-backed upper bound. Literature novelty is not claimed.
family generalized bicycle (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

[[72,10,9]] — published 2BGA code on SmallGroup(36,2)

Direction & hypothesis

This is a literature reconstruction for the weight-8 × unrestricted cell. Lin--Pryadko's arXiv:2306.16400v1 database provides explicit finite group and support data for connected binary 2BGA codes that are not all represented on the challenge board.

What was searched

The authors' public database was audited at github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c. This row is SmallGroup(36,2) with database supports a=[2,3,16] and b=[6,17,21], using 1-based GAP positions excluding the identity. The identity was added and the group was rebuilt with GAP regular representations.

Evidence trail

The reconstructed matrices have n=72, k=10, maximum check weight 8, and CSS commutation. The repository submission builder generated persisted X/Z logical witnesses of weight 9. verify/validate_candidate.py returned passed: true, no lighter logical in 5,380 RIS trials, no exact duplicate, no WL-equivalent duplicate, and board_advancing: true for weight-8 × unrestricted.

The paper reports d=9; this submission records only the witness-backed upper bound d <= 9 with upper_bound confidence. Literature novelty is not claimed: this is a published baseline reconstruction.

Dead ends

No search beyond reconstruction was performed. The row was selected because its parameter set was absent from the board comparison and it passed the computed Pareto check.

Tools

GAP for SmallGroup enumeration and the repository's research/kit/group_algebra.py, submit.py, and trusted validator. No changes to verify/ were made.

Reproduction

Use GAP SmallGroup(36,2) and add the identity to the database supports: a=1+g2+g3+g16, b=1+g6+g17+g21, with the database's 1-based element positions. Construct H_X=[L(a)|R(b)] and H_Z=[R(b)^T|L(a)^T] over GF(2), then run the repository validator.

Parity checks

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