[[152,4,18]] · QEC Challenge

[[152,4,18]] d ≤

152

kd²/n

8.526

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[2, 12, 26, 27, 28, 36, 61, 71, 76, 79, 81, 88, 105, 107, 108, 111, 115, 142]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[1, 7, 20, 21, 38, 61, 64, 75, 78, 110, 111, 117, 122, 123, 139, 146, 147, 151]

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)×76 (2,2)×76 (3,2)×76 (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 (2,2): 76 (2,6): 912 (2,8): 76 (2,10): 988 (3,2): 76 (3,4): 76 (3,6): 5244 (3,8): 2508 (3,10): 20748 (3,12): 6004 (3,14): 15352 (3,16): 988

trapping sets H_Z (1,2)×76 (2,2)×76 (3,2)×76 (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 (2,2): 76 (2,6): 912 (2,8): 76 (2,10): 988 (3,2): 76 (3,4): 76 (3,6): 5244 (3,8): 2508 (3,10): 20748 (3,12): 6004 (3,14): 15352 (3,16): 988

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_k4.txt, line 12. nonidentity supports a=[4], b=[2, 7, 10, 33, 53].

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

[[152,4,18]] — 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 = 8.53.

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_k4.txt, line 12. This row is SmallGroup(76,1) with nonidentity GAP supports a=[4], b=[2, 7, 10, 33, 53]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 18, 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=[4], b=[2, 7, 10, 33, 53] 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) = 4. 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, 71, 76, 80, 88, 105, 125, 150] [1, 5, 76, 77, 83, 106, 126, 151] [2, 74, 77, 78, 85, 108, 128, 147] [0, 3, 78, 79, 84, 101, 121, 151] [4, 8, 78, 80, 86, 103, 123, 149] [5, 9, 79, 80, 81, 102, 122, 147] [2, 6, 76, 81, 82, 104, 124, 149] [3, 7, 80, 82, 83, 97, 117, 148] [8, 12, 77, 82, 84, 99, 119, 150] [9, 13, 76, 84, 85, 98, 118, 143] [6, 10, 77, 79, 86, 100, 120, 145] [7, 11, 86, 87, 93, 113, 144, 151] [12, 16, 78, 81, 88, 95, 115, 146] [13, 17, 88, 89, 94, 114, 139, 147] [10, 14, 83, 90, 96, 116, 141, 149] [11, 15, 89, 90, 91, 109, 140, 148] [16, 20, 85, 91, 92, 111, 142, 150] [17, 21, 90, 92, 93, 110, 135, 143] [14, 18, 87, 92, 94, 112, 137, 145] [15, 19, 85, 94, 95, 105, 136, 144] [20, 24, 87, 89, 96, 107, 138, 146] [21, 25, 86, 96, 97, 106, 131, 139] [18, 22, 88, 91, 98, 108, 133, 141] [19, 23, 81, 98, 99, 101, 132, 140] [24, 28, 83, 93, 100, 103, 134, 142] [25, 29, 82, 100, 101, 102, 127, 135] [22, 26, 84, 95, 102, 104, 129, 137] [23, 27, 77, 97, 102, 103, 128, 136] [28, 32, 79, 97, 99, 104, 130, 138] [29, 33, 78, 98, 104, 105, 123, 131] [26, 30, 80, 99, 100, 106, 125, 133] [27, 31, 93, 106, 107, 124, 132, 149] [32, 36, 76, 95, 101, 108, 126, 134] [33, 37, 94, 108, 109, 119, 127, 150] [30, 34, 96, 103, 110, 121, 129, 151] [31, 35, 89, 110, 111, 120, 128, 145] [36, 40, 91, 105, 112, 122, 130, 147] [37, 41, 90, 112, 113, 115, 123, 146] [34, 38, 92, 107, 114, 117, 125, 148] [35, 39, 85, 114, 115, 116, 124, 141] [40, 44, 87, 109, 116, 118, 126, 143] [41, 45, 86, 111, 116, 117, 119, 142] [38, 42, 88, 111, 113, 118, 121, 144] [39, 43, 81, 112, 118, 119, 120, 137] [44, 48, 83, 113, 114, 120, 122, 139] [45, 49, 82, 107, 115, 120, 121, 138] [42, 46, 84, 109, 115, 117, 122, 140] [43, 47, 77, 108, 116, 122, 123, 133] [48, 52, 79, 110, 117, 118, 124, 135] [49, 53, 78, 103, 111, 124, 125, 134] [46, 50, 80, 105, 113, 119, 126, 136] [47, 51, 104, 112, 126, 127, 129, 149] [52, 56, 76, 106, 114, 121, 128, 131] [53, 57, 99, 107, 128, 129, 130, 150] [50, 54, 101, 109, 123, 130, 132, 151] [51, 55, 100, 108, 125, 130, 131, 145] [56, 60, 102, 110, 125, 127, 132, 147] [57, 61, 95, 103, 126, 132, 133, 146] [54, 58, 97, 105, 127, 128, 134, 148] [55, 59, 96, 104, 121, 134, 135, 141] [60, 64, 98, 106, 123, 129, 136, 143] [61, 65, 91, 99, 122, 136, 137, 142] [58, 62, 93, 101, 124, 131, 138, 144] [59, 63, 92, 100, 117, 137, 138, 139] [64, 68, 94, 102, 119, 133, 139, 140] [65, 69, 87, 95, 118, 138, 140, 141] [62, 66, 89, 97, 120, 135, 140, 142] [63, 67, 88, 96, 113, 133, 142, 143] [68, 72, 90, 98, 115, 135, 137, 144] [69, 73, 83, 91, 114, 134, 144, 145] [66, 70, 85, 93, 116, 136, 139, 146] [67, 71, 84, 92, 109, 129, 146, 147] [72, 75, 86, 94, 111, 131, 141, 148] [1, 73, 79, 87, 110, 130, 148, 149] [70, 74, 81, 89, 112, 132, 143, 150] [4, 75, 82, 90, 107, 127, 145, 151]

H_Z (76 checks, sparse supports)

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

Code ID 152-4-18 · download JSON · raw on GitHub