[[152,58,2]] · QEC Challenge

[[152,58,2]] d =

152

kd²/n

1.526

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)

[83, 88]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[57, 58]

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 4 · H_Z 4

trapping sets H_X (1,4)×152 (2,0)×114 (3,0)×16 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 152 (2,0): 114 (2,4): 140 (2,6): 1392 (3,0): 16 (3,4): 2984 (3,6): 1600 (3,8): 17520 (3,10): 1024

trapping sets H_Z (1,4)×152 (2,0)×114 (3,0)×16 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 152 (2,0): 114 (2,4): 140 (2,6): 1392 (3,0): 16 (3,4): 2984 (3,6): 1600 (3,8): 17520 (3,10): 1024

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_wt8wtL4_order76_k58.txt, line 1. nonidentity supports a=[2, 3, 5], b=[2, 4, 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

[[152,58,2]] — 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 = 1.53.

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_wt8wtL4_order76_k58.txt, line 1. This row is SmallGroup(76,1) with nonidentity GAP supports a=[2, 3, 5], b=[2, 4, 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=58), 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=[2, 3, 5], b=[2, 4, 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 = 152 - rank(HX) - rank(HZ) = 58. 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, 1, 2, 4, 76, 80, 81, 147] [0, 1, 2, 4, 76, 77, 82, 149] [0, 1, 2, 4, 77, 78, 84, 150] [3, 5, 6, 8, 76, 77, 79, 151] [0, 1, 2, 4, 78, 79, 80, 151] [3, 5, 6, 8, 77, 78, 81, 147] [3, 5, 6, 8, 78, 80, 82, 149] [7, 9, 10, 12, 79, 83, 148, 149] [3, 5, 6, 8, 76, 80, 84, 150] [7, 9, 10, 12, 81, 85, 143, 150] [7, 9, 10, 12, 82, 86, 145, 151] [11, 13, 14, 16, 83, 87, 144, 145] [7, 9, 10, 12, 84, 88, 146, 147] [11, 13, 14, 16, 85, 89, 139, 146] [11, 13, 14, 16, 86, 90, 141, 148] [15, 17, 18, 20, 87, 91, 140, 141] [11, 13, 14, 16, 88, 92, 142, 143] [15, 17, 18, 20, 89, 93, 135, 142] [15, 17, 18, 20, 90, 94, 137, 144] [19, 21, 22, 24, 91, 95, 136, 137] [15, 17, 18, 20, 92, 96, 138, 139] [19, 21, 22, 24, 93, 97, 131, 138] [19, 21, 22, 24, 94, 98, 133, 140] [23, 25, 26, 28, 95, 99, 132, 133] [19, 21, 22, 24, 96, 100, 134, 135] [23, 25, 26, 28, 97, 101, 127, 134] [23, 25, 26, 28, 98, 102, 129, 136] [27, 29, 30, 32, 99, 103, 128, 129] [23, 25, 26, 28, 100, 104, 130, 131] [27, 29, 30, 32, 101, 105, 123, 130] [27, 29, 30, 32, 102, 106, 125, 132] [31, 33, 34, 36, 103, 107, 124, 125] [27, 29, 30, 32, 104, 108, 126, 127] [31, 33, 34, 36, 105, 109, 119, 126] [31, 33, 34, 36, 106, 110, 121, 128] [35, 37, 38, 40, 107, 111, 120, 121] [31, 33, 34, 36, 108, 112, 122, 123] [35, 37, 38, 40, 109, 113, 115, 122] [35, 37, 38, 40, 110, 114, 117, 124] [39, 41, 42, 44, 111, 115, 116, 117] [35, 37, 38, 40, 112, 116, 118, 119] [39, 41, 42, 44, 111, 113, 117, 118] [39, 41, 42, 44, 113, 114, 118, 120] [43, 45, 46, 48, 112, 113, 115, 119] [39, 41, 42, 44, 114, 115, 116, 120] [43, 45, 46, 48, 107, 114, 117, 121] [43, 45, 46, 48, 109, 116, 118, 122] [47, 49, 50, 52, 108, 109, 119, 123] [43, 45, 46, 48, 110, 111, 120, 124] [47, 49, 50, 52, 103, 110, 121, 125] [47, 49, 50, 52, 105, 112, 122, 126] [51, 53, 54, 56, 104, 105, 123, 127] [47, 49, 50, 52, 106, 107, 124, 128] [51, 53, 54, 56, 99, 106, 125, 129] [51, 53, 54, 56, 101, 108, 126, 130] [55, 57, 58, 60, 100, 101, 127, 131] [51, 53, 54, 56, 102, 103, 128, 132] [55, 57, 58, 60, 95, 102, 129, 133] [55, 57, 58, 60, 97, 104, 130, 134] [59, 61, 62, 64, 96, 97, 131, 135] [55, 57, 58, 60, 98, 99, 132, 136] [59, 61, 62, 64, 91, 98, 133, 137] [59, 61, 62, 64, 93, 100, 134, 138] [63, 65, 66, 68, 92, 93, 135, 139] [59, 61, 62, 64, 94, 95, 136, 140] [63, 65, 66, 68, 87, 94, 137, 141] [63, 65, 66, 68, 89, 96, 138, 142] [67, 69, 70, 72, 88, 89, 139, 143] [63, 65, 66, 68, 90, 91, 140, 144] [67, 69, 70, 72, 83, 90, 141, 145] [67, 69, 70, 72, 85, 92, 142, 146] [71, 73, 74, 75, 84, 85, 143, 147] [67, 69, 70, 72, 86, 87, 144, 148] [71, 73, 74, 75, 79, 86, 145, 149] [71, 73, 74, 75, 81, 88, 146, 150] [71, 73, 74, 75, 82, 83, 148, 151]

H_Z (76 checks, sparse supports)

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

Code ID 152-58-2 · download JSON · raw on GitHub