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[[106,2,15]] d ≤
n
106
k
2
d
15
kd²/n
4.245
w
8
X/Z
1
g
0.0021
r
6.7082
layers
2
swaps
652

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Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[38, 39, 40, 41, 42, 73, 75, 77, 78, 84, 85, 87, 89, 90, 95]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[0, 2, 10, 12, 18, 38, 39, 48, 49, 50, 53, 91, 97, 99, 105]
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 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 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×53 (2,2)×53 (3,2)×53 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 53 (1,6): 53 (2,2): 53 (2,4): 53 (2,6): 583 (2,8): 106 (2,10): 424 (3,2): 53 (3,4): 477 (3,6): 3392 (3,8): 3233 (3,10): 8904 (3,12): 3127 (3,14): 3975 (3,16): 159
trapping sets H_Z (1,2)×53 (2,2)×53 (3,2)×53 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 53 (1,6): 53 (2,2): 53 (2,4): 53 (2,6): 583 (2,8): 106 (2,10): 424 (3,2): 53 (3,4): 477 (3,6): 3392 (3,8): 3233 (3,10): 8904 (3,12): 3127 (3,14): 3975 (3,16): 159
witness diameter X 6.3246 · Z 7.6158 (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 (53)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 652 nearest-neighbor SWAPs per round in total, at most 11 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 novelty not audited
construction Generalized bicycle code (arXiv:2306.16400, Renyu Wang & Leonid P. Pryadko) from github.com/QEC-pages/GB-codes, gb-codes.zip, codes/GB_106_w8_X.mtx and _Z.mtx.
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-19
notes Literature reproduction of a published generalized bicycle code (arXiv:2306.16400, Renyu Wang & Leonid P. Pryadko), from github.com/QEC-pages/GB-codes gb-codes.zip. Checked against the live board: not equivalent to any existing entry.
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

[[106,2,15]] — generalized bicycle code on Z_{ell} (Wang–Pryadko)

Direction & hypothesis

Target: the unrestricted weight-8 frontier for hackathon issue #1155. This is one of the original generalized bicycle codes of Renyu Wang and Leonid P. Pryadko (arXiv:2306.16400, "Distance bounds for generalized bicycle codes"). It was absent from the board at the time of submission, and the trusted validator reports that it advances the weight-8 unrestricted cell. Efficiency kd²/n = 4.245.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts. From github.com/QEC-pages/GB-codes @ eb3113d, gb-codes.zip, the codes/GB_106_w8_X.mtx and codes/GB_106_w8_Z.mtx pair. These are the actual H_X and H_Z parity-check matrices shipped by the authors; no reconstruction from group parameters was needed. The code is a two-block generalized bicycle code over the circulant ring Z_{ell} with ell = 53.

Evidence trail

The trusted candidate gate returned passed: true — structural verification passed (CSS commutation, connectivity, k = 2), and the random-seed refutation found no lighter logical than the claimed distance. Distance is a witness-backed upper bound d ≤ 15, not an exact certificate. The published distance is consistent with this run.

Dead ends

The same repo ships 72 GB codes; most are dominated by existing board entries or already submitted (the 2BGA-codes parameter mining covered a different parameterization). This (n,k) = (106,2) pair survived the board-dominance screen, the within-pool Pareto filter, and the trusted gate.

Tools

DeepSeek V4 Flash 0731 (provenance.model), Zed coding agent; NumPy, the repository submission builder (research/kit/submit.py) and the trusted validator (verify/validate_candidate.py). Public artifacts fetched by immutable commit URL and treated as data. Bounded local CPU run, no paid compute.

Reproduction

Read the two Matrix Market files GB_106_w8_X.mtx and GB_106_w8_Z.mtx from github.com/QEC-pages/GB-codes @ eb3113d, gb-codes.zip. Each is a 0/1 coordinate matrix: header line nchecks n nnz, then i j 1 entries (1-based). Build dense H_X, H_Z, verify CSS commutation H_X·H_Zᵀ = 0, and pass both to the repository submission builder with confidence upper_bound. Save the returned witnesses immediately and run the default trusted candidate gate. Do not replace the returned distance with the paper's number.

Parity checks

X-checks 53 (max weight 8) · Z-checks 53 (max weight 8)
H_X (53 checks, sparse supports)
[0, 52, 53, 87, 93, 94, 100, 104] [0, 1, 54, 88, 94, 95, 101, 105] [1, 2, 53, 55, 89, 95, 96, 102] [2, 3, 54, 56, 90, 96, 97, 103] [3, 4, 55, 57, 91, 97, 98, 104] [4, 5, 56, 58, 92, 98, 99, 105] [5, 6, 53, 57, 59, 93, 99, 100] [6, 7, 54, 58, 60, 94, 100, 101] [7, 8, 55, 59, 61, 95, 101, 102] [8, 9, 56, 60, 62, 96, 102, 103] [9, 10, 57, 61, 63, 97, 103, 104] [10, 11, 58, 62, 64, 98, 104, 105] [11, 12, 53, 59, 63, 65, 99, 105] [12, 13, 53, 54, 60, 64, 66, 100] [13, 14, 54, 55, 61, 65, 67, 101] [14, 15, 55, 56, 62, 66, 68, 102] [15, 16, 56, 57, 63, 67, 69, 103] [16, 17, 57, 58, 64, 68, 70, 104] [17, 18, 58, 59, 65, 69, 71, 105] [18, 19, 53, 59, 60, 66, 70, 72] [19, 20, 54, 60, 61, 67, 71, 73] [20, 21, 55, 61, 62, 68, 72, 74] [21, 22, 56, 62, 63, 69, 73, 75] [22, 23, 57, 63, 64, 70, 74, 76] [23, 24, 58, 64, 65, 71, 75, 77] [24, 25, 59, 65, 66, 72, 76, 78] [25, 26, 60, 66, 67, 73, 77, 79] [26, 27, 61, 67, 68, 74, 78, 80] [27, 28, 62, 68, 69, 75, 79, 81] [28, 29, 63, 69, 70, 76, 80, 82] [29, 30, 64, 70, 71, 77, 81, 83] [30, 31, 65, 71, 72, 78, 82, 84] [31, 32, 66, 72, 73, 79, 83, 85] [32, 33, 67, 73, 74, 80, 84, 86] [33, 34, 68, 74, 75, 81, 85, 87] [34, 35, 69, 75, 76, 82, 86, 88] [35, 36, 70, 76, 77, 83, 87, 89] [36, 37, 71, 77, 78, 84, 88, 90] [37, 38, 72, 78, 79, 85, 89, 91] [38, 39, 73, 79, 80, 86, 90, 92] [39, 40, 74, 80, 81, 87, 91, 93] [40, 41, 75, 81, 82, 88, 92, 94] [41, 42, 76, 82, 83, 89, 93, 95] [42, 43, 77, 83, 84, 90, 94, 96] [43, 44, 78, 84, 85, 91, 95, 97] [44, 45, 79, 85, 86, 92, 96, 98] [45, 46, 80, 86, 87, 93, 97, 99] [46, 47, 81, 87, 88, 94, 98, 100] [47, 48, 82, 88, 89, 95, 99, 101] [48, 49, 83, 89, 90, 96, 100, 102] [49, 50, 84, 90, 91, 97, 101, 103] [50, 51, 85, 91, 92, 98, 102, 104] [51, 52, 86, 92, 93, 99, 103, 105]
H_Z (53 checks, sparse supports)
[0, 2, 6, 12, 13, 19, 53, 54] [1, 3, 7, 13, 14, 20, 54, 55] [2, 4, 8, 14, 15, 21, 55, 56] [3, 5, 9, 15, 16, 22, 56, 57] [4, 6, 10, 16, 17, 23, 57, 58] [5, 7, 11, 17, 18, 24, 58, 59] [6, 8, 12, 18, 19, 25, 59, 60] [7, 9, 13, 19, 20, 26, 60, 61] [8, 10, 14, 20, 21, 27, 61, 62] [9, 11, 15, 21, 22, 28, 62, 63] [10, 12, 16, 22, 23, 29, 63, 64] [11, 13, 17, 23, 24, 30, 64, 65] [12, 14, 18, 24, 25, 31, 65, 66] [13, 15, 19, 25, 26, 32, 66, 67] [14, 16, 20, 26, 27, 33, 67, 68] [15, 17, 21, 27, 28, 34, 68, 69] [16, 18, 22, 28, 29, 35, 69, 70] [17, 19, 23, 29, 30, 36, 70, 71] [18, 20, 24, 30, 31, 37, 71, 72] [19, 21, 25, 31, 32, 38, 72, 73] [20, 22, 26, 32, 33, 39, 73, 74] [21, 23, 27, 33, 34, 40, 74, 75] [22, 24, 28, 34, 35, 41, 75, 76] [23, 25, 29, 35, 36, 42, 76, 77] [24, 26, 30, 36, 37, 43, 77, 78] [25, 27, 31, 37, 38, 44, 78, 79] [26, 28, 32, 38, 39, 45, 79, 80] [27, 29, 33, 39, 40, 46, 80, 81] [28, 30, 34, 40, 41, 47, 81, 82] [29, 31, 35, 41, 42, 48, 82, 83] [30, 32, 36, 42, 43, 49, 83, 84] [31, 33, 37, 43, 44, 50, 84, 85] [32, 34, 38, 44, 45, 51, 85, 86] [33, 35, 39, 45, 46, 52, 86, 87] [0, 34, 36, 40, 46, 47, 87, 88] [1, 35, 37, 41, 47, 48, 88, 89] [2, 36, 38, 42, 48, 49, 89, 90] [3, 37, 39, 43, 49, 50, 90, 91] [4, 38, 40, 44, 50, 51, 91, 92] [5, 39, 41, 45, 51, 52, 92, 93] [0, 6, 40, 42, 46, 52, 93, 94] [0, 1, 7, 41, 43, 47, 94, 95] [1, 2, 8, 42, 44, 48, 95, 96] [2, 3, 9, 43, 45, 49, 96, 97] [3, 4, 10, 44, 46, 50, 97, 98] [4, 5, 11, 45, 47, 51, 98, 99] [5, 6, 12, 46, 48, 52, 99, 100] [0, 6, 7, 13, 47, 49, 100, 101] [1, 7, 8, 14, 48, 50, 101, 102] [2, 8, 9, 15, 49, 51, 102, 103] [3, 9, 10, 16, 50, 52, 103, 104] [0, 4, 10, 11, 17, 51, 104, 105] [1, 5, 11, 12, 18, 52, 53, 105]
Code ID 106-2-15 · download JSON · raw on GitHub