[[152,76,2]] · QEC Challenge

[[152,76,2]] d =

152

kd²/n

2.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[25, 28]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[89, 92]

certificate exact, d = 2 · scipy/HiGHS MILP

X: no logical < 2 exists; Z: no logical < 2 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)×76 (2,0)×76 (3,2)×1748 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 76 (1,6): 76 (2,0): 76 (2,4): 608 (2,8): 152 (3,2): 1748 (3,6): 3268 (3,10): 608

trapping sets H_Z (1,2)×76 (2,0)×76 (3,2)×1748 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 76 (1,6): 76 (2,0): 76 (2,4): 608 (2,8): 152 (3,2): 1748 (3,6): 3268 (3,10): 608

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(76,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order76_k76.txt, line 1. nonidentity supports a=[3], b=[2, 3, 4, 5, 7].

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

[[152,76,2]] — two-block group-algebra code on SmallGroup(76,1)

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 = 2.00.

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_order76_k76.txt, line 1. This row is SmallGroup(76,1) with nonidentity GAP supports a=[3], b=[2, 3, 4, 5, 7]. 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=76), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, 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(76,1), 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, 5, 7] 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 = 152 - rank(HX) - rank(HZ) = 76. 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 76 (max weight 8) · Z-checks 76 (max weight 8)

H_X (76 checks, sparse supports)

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

H_Z (76 checks, sparse supports)

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

Code ID 152-76-2 · download JSON · raw on GitHub