[[126,42,3]] · QEC Challenge

[[126,42,3]] d =

126

kd²/n

3.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[44, 52, 58]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[38, 46, 53]

certificate exact, d = 3 · scipy/HiGHS MILP

X: no logical < 3 exists; Z: no logical < 3 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 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)

trapping sets H_X (1,2)×63 (2,2)×63 (3,0)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 63 (1,6): 63 (2,2): 63 (2,4): 189 (2,6): 441 (2,8): 189 (2,10): 378 (3,0): 42 (3,2): 189 (3,4): 1134 (3,6): 2268 (3,8): 3906 (3,10): 5859 (3,12): 2772 (3,14): 2961

trapping sets H_Z (1,2)×63 (2,2)×63 (3,0)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 63 (1,6): 63 (2,2): 63 (2,4): 189 (2,6): 441 (2,8): 189 (2,10): 378 (3,0): 42 (3,2): 189 (3,4): 1134 (3,6): 2268 (3,8): 3906 (3,10): 5859 (3,12): 2772 (3,14): 2961

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(63,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order63_k42.txt, line 1. nonidentity supports a=[3], b=[2, 3, 4, 6, 9].

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-19

notes Literature reproduction of a published 2BGA code (arXiv:2306.16400; github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c). Checked against the live board: not equivalent to any existing entry.

family generalized bicycle (a tag, not a ranking)

locality unrestricted (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

[[126,42,3]] — two-block group-algebra code on SmallGroup(63,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 3.00.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order63_k42.txt, line 1. This row is SmallGroup(63,1) with nonidentity GAP supports a=[3], b=[2, 3, 4, 6, 9]. Reconstructed via GAP (Cayley table) and the repository's group-algebra constructor.

Evidence trail

The trusted candidate gate returned passed: true, structural verification passed (CSS, connectivity, k=42), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 3, not an exact certificate. The published distance is consistent with this run.

Dead ends

None for this candidate. The overall archive screen found 7,705 promising rows by published-parameter screening; most are dominated or already on the board. This row survived the screen and the gate.

Tools

DeepSeek V4 Flash 0731 (provenance.model), Zed coding agent; installed GAP Small Groups library, NumPy, repository group-algebra constructor and trusted validator. Public artifacts fetched by immutable commit URL and treated as data. Bounded local CPU run, no paid compute.

Reproduction

Use GAP g := SmallGroup(63,1), e := Elements(g). Construct the zero-based Cayley table mul[i,j] = Position(e,e[i+1]*e[j+1])-1. The published nonidentity supports are a=[3], b=[2, 3, 4, 6, 9] in GAP's one-based indexing. Add identity and convert to zero-based. Build HX=[L(a)|R(b)], HZ=[R(b)^T|L(a)^T] with left/right regular representations. Verify CSS; k = 126 - rank(HX) - rank(HZ) = 42. Pass 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 63 (max weight 8) · Z-checks 63 (max weight 8)

H_X (63 checks, sparse supports)

[0, 7, 63, 70, 73, 81, 107, 121] [1, 12, 63, 64, 70, 75, 113, 124] [0, 2, 63, 65, 67, 81, 107, 115] [3, 15, 63, 66, 70, 78, 109, 116] [4, 18, 64, 67, 75, 81, 118, 125] [1, 5, 63, 64, 65, 68, 113, 120] [6, 21, 64, 69, 75, 84, 89, 105] [2, 7, 65, 67, 70, 73, 115, 121] [3, 8, 63, 65, 66, 71, 101, 116] [9, 24, 66, 72, 78, 82, 87, 90] [4, 10, 64, 67, 68, 73, 118, 123] [11, 27, 67, 74, 81, 90, 95, 111] [5, 12, 65, 68, 70, 75, 120, 124] [6, 13, 64, 68, 69, 76, 89, 97] [14, 30, 66, 69, 77, 78, 84, 93] [8, 15, 65, 70, 71, 78, 101, 109] [9, 16, 66, 71, 72, 74, 79, 90] [17, 33, 72, 80, 87, 96, 117, 122] [10, 18, 68, 73, 75, 81, 123, 125] [11, 19, 67, 73, 74, 82, 95, 103] [20, 36, 69, 74, 83, 84, 90, 99] [13, 21, 68, 75, 76, 84, 97, 105] [14, 22, 66, 69, 71, 76, 77, 85] [23, 39, 77, 86, 93, 98, 102, 114] [16, 24, 71, 74, 78, 79, 82, 87] [17, 25, 72, 79, 80, 88, 110, 122] [26, 42, 80, 89, 91, 96, 99, 105] [19, 27, 73, 81, 82, 90, 103, 111] [20, 28, 69, 74, 76, 82, 83, 91] [29, 45, 83, 92, 99, 104, 108, 119] [22, 30, 71, 76, 78, 84, 85, 93] [23, 31, 77, 85, 86, 94, 98, 106] [32, 48, 72, 86, 87, 95, 102, 111] [25, 33, 79, 87, 88, 96, 110, 117] [26, 34, 80, 83, 88, 89, 97, 99] [35, 51, 89, 98, 105, 114, 123, 125] [28, 36, 76, 82, 84, 90, 91, 99] [29, 37, 83, 91, 92, 100, 104, 112] [38, 53, 77, 92, 93, 101, 108, 116] [31, 39, 85, 93, 94, 102, 106, 114] [32, 40, 72, 79, 86, 94, 95, 103] [41, 56, 95, 104, 107, 111, 119, 121] [34, 42, 83, 88, 91, 96, 97, 105] [35, 43, 89, 97, 98, 106, 118, 125] [44, 58, 98, 100, 107, 108, 114, 121] [37, 45, 91, 99, 100, 108, 112, 119] [38, 46, 77, 85, 92, 100, 101, 109] [47, 59, 101, 110, 113, 116, 122, 124] [40, 48, 79, 87, 94, 102, 103, 111] [41, 49, 95, 103, 104, 107, 112, 115] [50, 61, 80, 96, 104, 113, 119, 124] [43, 51, 97, 105, 106, 114, 118, 123] [44, 52, 92, 98, 106, 107, 108, 115] [46, 53, 85, 93, 100, 108, 109, 116] [47, 54, 101, 109, 110, 113, 117, 120] [55, 62, 86, 102, 110, 118, 122, 125] [49, 56, 103, 111, 112, 115, 119, 121] [50, 57, 80, 88, 104, 112, 113, 120] [52, 58, 92, 100, 106, 114, 115, 121] [54, 59, 109, 116, 117, 120, 122, 124] [55, 60, 86, 94, 110, 117, 118, 123] [57, 61, 88, 96, 112, 119, 120, 124] [60, 62, 94, 102, 117, 122, 123, 125]

H_Z (63 checks, sparse supports)

[0, 1, 2, 3, 5, 8, 63, 65] [1, 4, 5, 6, 10, 13, 64, 68] [2, 5, 7, 8, 12, 15, 65, 70] [3, 8, 9, 14, 16, 22, 66, 71] [2, 4, 7, 10, 11, 19, 67, 73] [5, 10, 12, 13, 18, 21, 68, 75] [6, 13, 14, 20, 22, 28, 69, 76] [0, 1, 3, 7, 12, 15, 63, 70] [8, 15, 16, 22, 24, 30, 71, 78] [9, 16, 17, 25, 32, 40, 72, 79] [0, 7, 10, 18, 19, 27, 73, 81] [11, 16, 19, 20, 24, 28, 74, 82] [1, 4, 6, 12, 18, 21, 64, 75] [13, 21, 22, 28, 30, 36, 76, 84] [14, 22, 23, 31, 38, 46, 77, 85] [3, 9, 14, 15, 24, 30, 66, 78] [16, 24, 25, 33, 40, 48, 79, 87] [17, 25, 26, 34, 50, 57, 80, 88] [0, 2, 4, 11, 18, 27, 67, 81] [9, 19, 24, 27, 28, 36, 82, 90] [20, 28, 29, 34, 37, 42, 83, 91] [6, 14, 20, 21, 30, 36, 69, 84] [22, 30, 31, 39, 46, 53, 85, 93] [23, 31, 32, 40, 55, 60, 86, 94] [9, 17, 24, 32, 33, 48, 72, 87] [25, 33, 34, 42, 57, 61, 88, 96] [6, 13, 26, 34, 35, 43, 89, 97] [9, 11, 16, 20, 27, 36, 74, 90] [26, 28, 36, 37, 42, 45, 91, 99] [29, 37, 38, 46, 52, 58, 92, 100] [14, 23, 30, 38, 39, 53, 77, 93] [31, 39, 40, 48, 60, 62, 94, 102] [11, 19, 32, 40, 41, 49, 95, 103] [17, 26, 33, 42, 50, 61, 80, 96] [13, 21, 34, 42, 43, 51, 97, 105] [23, 31, 35, 43, 44, 52, 98, 106] [20, 26, 29, 34, 36, 45, 83, 99] [37, 44, 45, 46, 53, 58, 100, 108] [8, 15, 38, 46, 47, 54, 101, 109] [23, 32, 39, 48, 55, 62, 86, 102] [19, 27, 40, 48, 49, 56, 103, 111] [29, 37, 41, 49, 50, 57, 104, 112] [6, 21, 26, 35, 42, 51, 89, 105] [31, 39, 43, 51, 52, 58, 106, 114] [0, 2, 41, 44, 49, 52, 107, 115] [29, 38, 44, 45, 52, 53, 92, 108] [3, 15, 46, 53, 54, 59, 109, 116] [25, 33, 47, 54, 55, 60, 110, 117] [11, 27, 32, 41, 48, 56, 95, 111] [37, 45, 49, 56, 57, 61, 112, 119] [1, 5, 47, 50, 54, 57, 113, 120] [23, 35, 39, 44, 51, 58, 98, 114] [2, 7, 49, 52, 56, 58, 115, 121] [3, 8, 38, 47, 53, 59, 101, 116] [17, 33, 54, 59, 60, 62, 117, 122] [4, 10, 43, 51, 55, 60, 118, 123] [29, 41, 45, 50, 56, 61, 104, 119] [5, 12, 54, 57, 59, 61, 120, 124] [0, 7, 41, 44, 56, 58, 107, 121] [17, 25, 47, 55, 59, 62, 110, 122] [10, 18, 35, 51, 60, 62, 123, 125] [1, 12, 47, 50, 59, 61, 113, 124] [4, 18, 35, 43, 55, 62, 118, 125]

Code ID 126-42-3 · download JSON · raw on GitHub