[[144,8,14]] · QEC Challenge

[[144,8,14]] d ≤

144

kd²/n

10.889

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 14 · witness weight 14 (claimed upper_bound)

witness operator (support, 14 qubits)

[33, 45, 49, 53, 73, 74, 75, 76, 93, 94, 100, 106, 135, 138]

d_Z 14 · witness weight 14 (claimed upper_bound)

witness operator (support, 14 qubits)

[2, 20, 33, 35, 44, 45, 46, 63, 65, 69, 76, 93, 123, 142]

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 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×72 (2,3)×36 (3,3)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 72 (1,4): 72 (2,3): 36 (2,4): 252 (2,5): 792 (2,6): 360 (3,3): 72 (3,4): 396 (3,5): 1584 (3,6): 6840 (3,7): 8748 (3,8): 3096 (3,9): 900 (3,10): 144

trapping sets H_Z (1,3)×72 (2,3)×36 (3,3)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 72 (1,4): 72 (2,3): 36 (2,4): 252 (2,5): 792 (2,6): 360 (3,3): 72 (3,4): 396 (3,5): 1584 (3,6): 6840 (3,7): 8748 (3,8): 3096 (3,9): 900 (3,10): 144

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-18

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

[[144,8,14]] — two-block group-algebra code on SmallGroup(72,8)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 10.89.

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_wt7wtL3_order72_k8.txt, line 31. This row is SmallGroup(72,8) with nonidentity GAP supports a=[12, 47], b=[2, 13, 27]. 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=8), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 14, not an exact certificate. The published distance (d=14) 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

Union Alpha (maker currently anonymous), 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(72,8), 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=[12, 47] and b=[2, 13, 27] 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 = 144 - rank(HX) - rank(HZ) = 8. 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 72 (max weight 7) · Z-checks 72 (max weight 7)

H_X (72 checks, sparse supports)

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

H_Z (72 checks, sparse supports)

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

Code ID 144-8-14 · download JSON · raw on GitHub