[[168,84,2]] · QEC Challenge

[[168,84,2]] d =

168

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)

[117, 128]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[23, 34]

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)×84 (2,0)×84 (3,2)×1596 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 84 (1,6): 84 (2,0): 84 (2,4): 588 (2,8): 336 (3,2): 1596 (3,6): 4480 (3,10): 2352

trapping sets H_Z (1,2)×84 (2,0)×84 (3,2)×1596 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 84 (1,6): 84 (2,0): 84 (2,4): 588 (2,8): 336 (3,2): 1596 (3,6): 4480 (3,10): 2352

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(84,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order84_k84.txt, line 1. nonidentity supports a=[4], b=[2, 4, 7, 11, 21].

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

[[168,84,2]] — two-block group-algebra code on SmallGroup(84,1)

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_order84_k84.txt, line 1. This row is SmallGroup(84,1) with nonidentity GAP supports a=[4], b=[2, 4, 7, 11, 21]. 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=84), 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(84,1), 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, 4, 7, 11, 21] 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 = 168 - rank(HX) - rank(HZ) = 84. 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 84 (max weight 8) · Z-checks 84 (max weight 8)

H_X (84 checks, sparse supports)

[0, 3, 84, 85, 87, 90, 151, 160] [1, 6, 84, 85, 87, 90, 156, 163] [2, 9, 86, 89, 93, 98, 131, 142] [0, 3, 84, 85, 87, 90, 151, 160] [4, 11, 88, 92, 95, 102, 149, 159] [5, 14, 86, 89, 93, 98, 137, 148] [1, 6, 84, 85, 87, 90, 156, 163] [7, 16, 91, 97, 100, 108, 143, 154] [8, 18, 92, 97, 102, 108, 141, 152] [2, 9, 86, 89, 93, 98, 131, 142] [10, 20, 84, 87, 94, 104, 158, 165] [4, 11, 88, 92, 95, 102, 149, 159] [12, 22, 96, 106, 115, 126, 137, 148] [13, 24, 92, 97, 102, 108, 147, 157] [5, 14, 86, 89, 93, 98, 137, 148] [15, 26, 85, 90, 99, 110, 153, 162] [7, 16, 91, 97, 100, 108, 143, 154] [17, 28, 101, 112, 121, 131, 132, 142] [8, 18, 92, 97, 102, 108, 141, 152] [19, 30, 86, 93, 103, 114, 164, 167] [10, 20, 84, 87, 94, 104, 158, 165] [21, 32, 96, 105, 106, 116, 147, 157] [12, 22, 96, 106, 115, 126, 137, 148] [23, 34, 107, 118, 125, 136, 139, 150] [13, 24, 92, 97, 102, 108, 147, 157] [25, 36, 89, 98, 109, 120, 161, 166] [15, 26, 85, 90, 99, 110, 153, 162] [27, 38, 101, 111, 112, 122, 141, 152] [17, 28, 101, 112, 121, 131, 132, 142] [29, 40, 113, 119, 124, 130, 145, 155] [19, 30, 86, 93, 103, 114, 164, 167] [31, 42, 105, 115, 116, 126, 156, 163] [21, 32, 96, 105, 106, 116, 147, 157] [33, 44, 117, 119, 128, 130, 135, 146] [23, 34, 107, 118, 125, 136, 139, 150] [35, 46, 113, 119, 124, 130, 161, 166] [25, 36, 89, 98, 109, 120, 161, 166] [37, 48, 111, 121, 122, 132, 151, 160] [27, 38, 101, 111, 112, 122, 141, 152] [39, 50, 123, 125, 129, 134, 136, 140] [29, 40, 113, 119, 124, 130, 145, 155] [41, 52, 107, 118, 125, 136, 164, 167] [31, 42, 105, 115, 116, 126, 156, 163] [43, 54, 127, 129, 138, 140, 145, 155] [33, 44, 117, 119, 128, 130, 135, 146] [45, 56, 123, 129, 134, 140, 143, 154] [35, 46, 113, 119, 124, 130, 161, 166] [47, 58, 101, 103, 112, 114, 131, 142] [37, 48, 111, 121, 122, 132, 151, 160] [49, 60, 133, 135, 139, 144, 146, 150] [39, 50, 123, 125, 129, 134, 136, 140] [51, 62, 117, 128, 135, 146, 149, 159] [41, 52, 107, 118, 125, 136, 164, 167] [53, 64, 96, 106, 109, 120, 137, 148] [43, 54, 127, 129, 138, 140, 145, 155] [55, 66, 133, 139, 144, 150, 153, 162] [45, 56, 123, 129, 134, 140, 143, 154] [57, 68, 88, 95, 111, 122, 141, 152] [47, 58, 101, 103, 112, 114, 131, 142] [59, 70, 91, 100, 127, 138, 143, 154] [49, 60, 133, 135, 139, 144, 146, 150] [61, 71, 127, 138, 145, 155, 158, 165] [51, 62, 117, 128, 135, 146, 149, 159] [63, 73, 91, 100, 105, 116, 147, 157] [53, 64, 96, 106, 109, 120, 137, 148] [65, 75, 88, 95, 133, 144, 149, 159] [55, 66, 133, 139, 144, 150, 153, 162] [67, 76, 94, 104, 121, 132, 151, 160] [57, 68, 88, 95, 111, 122, 141, 152] [69, 78, 99, 107, 110, 118, 153, 162] [59, 70, 91, 100, 127, 138, 143, 154] [61, 71, 127, 138, 145, 155, 158, 165] [72, 79, 99, 110, 115, 126, 156, 163] [63, 73, 91, 100, 105, 116, 147, 157] [74, 81, 94, 104, 113, 124, 158, 165] [65, 75, 88, 95, 133, 144, 149, 159] [67, 76, 94, 104, 121, 132, 151, 160] [77, 82, 109, 117, 120, 128, 161, 166] [69, 78, 99, 107, 110, 118, 153, 162] [72, 79, 99, 110, 115, 126, 156, 163] [80, 83, 103, 114, 123, 134, 164, 167] [74, 81, 94, 104, 113, 124, 158, 165] [77, 82, 109, 117, 120, 128, 161, 166] [80, 83, 103, 114, 123, 134, 164, 167]

H_Z (84 checks, sparse supports)

[0, 1, 3, 6, 10, 20, 84, 87] [0, 1, 3, 6, 15, 26, 85, 90] [2, 5, 9, 14, 19, 30, 86, 93] [0, 1, 3, 6, 10, 20, 84, 87] [4, 11, 57, 65, 68, 75, 88, 95] [2, 5, 9, 14, 25, 36, 89, 98] [0, 1, 3, 6, 15, 26, 85, 90] [7, 16, 59, 63, 70, 73, 91, 100] [4, 8, 11, 13, 18, 24, 92, 102] [2, 5, 9, 14, 19, 30, 86, 93] [10, 20, 67, 74, 76, 81, 94, 104] [4, 11, 57, 65, 68, 75, 88, 95] [12, 21, 22, 32, 53, 64, 96, 106] [7, 8, 13, 16, 18, 24, 97, 108] [2, 5, 9, 14, 25, 36, 89, 98] [15, 26, 69, 72, 78, 79, 99, 110] [7, 16, 59, 63, 70, 73, 91, 100] [17, 27, 28, 38, 47, 58, 101, 112] [4, 8, 11, 13, 18, 24, 92, 102] [19, 30, 47, 58, 80, 83, 103, 114] [10, 20, 67, 74, 76, 81, 94, 104] [21, 31, 32, 42, 63, 73, 105, 116] [12, 21, 22, 32, 53, 64, 96, 106] [23, 34, 41, 52, 69, 78, 107, 118] [7, 8, 13, 16, 18, 24, 97, 108] [25, 36, 53, 64, 77, 82, 109, 120] [15, 26, 69, 72, 78, 79, 99, 110] [27, 37, 38, 48, 57, 68, 111, 122] [17, 27, 28, 38, 47, 58, 101, 112] [29, 35, 40, 46, 74, 81, 113, 124] [19, 30, 47, 58, 80, 83, 103, 114] [12, 22, 31, 42, 72, 79, 115, 126] [21, 31, 32, 42, 63, 73, 105, 116] [33, 44, 51, 62, 77, 82, 117, 128] [23, 34, 41, 52, 69, 78, 107, 118] [29, 33, 35, 40, 44, 46, 119, 130] [25, 36, 53, 64, 77, 82, 109, 120] [17, 28, 37, 48, 67, 76, 121, 132] [27, 37, 38, 48, 57, 68, 111, 122] [39, 45, 50, 56, 80, 83, 123, 134] [29, 35, 40, 46, 74, 81, 113, 124] [23, 34, 39, 41, 50, 52, 125, 136] [12, 22, 31, 42, 72, 79, 115, 126] [43, 54, 59, 61, 70, 71, 127, 138] [33, 44, 51, 62, 77, 82, 117, 128] [39, 43, 45, 50, 54, 56, 129, 140] [29, 33, 35, 40, 44, 46, 119, 130] [2, 9, 17, 28, 47, 58, 131, 142] [17, 28, 37, 48, 67, 76, 121, 132] [49, 55, 60, 65, 66, 75, 133, 144] [39, 45, 50, 56, 80, 83, 123, 134] [33, 44, 49, 51, 60, 62, 135, 146] [23, 34, 39, 41, 50, 52, 125, 136] [5, 12, 14, 22, 53, 64, 137, 148] [43, 54, 59, 61, 70, 71, 127, 138] [23, 34, 49, 55, 60, 66, 139, 150] [39, 43, 45, 50, 54, 56, 129, 140] [8, 18, 27, 38, 57, 68, 141, 152] [2, 9, 17, 28, 47, 58, 131, 142] [7, 16, 45, 56, 59, 70, 143, 154] [49, 55, 60, 65, 66, 75, 133, 144] [29, 40, 43, 54, 61, 71, 145, 155] [33, 44, 49, 51, 60, 62, 135, 146] [13, 21, 24, 32, 63, 73, 147, 157] [5, 12, 14, 22, 53, 64, 137, 148] [4, 11, 51, 62, 65, 75, 149, 159] [23, 34, 49, 55, 60, 66, 139, 150] [0, 3, 37, 48, 67, 76, 151, 160] [8, 18, 27, 38, 57, 68, 141, 152] [15, 26, 55, 66, 69, 78, 153, 162] [7, 16, 45, 56, 59, 70, 143, 154] [29, 40, 43, 54, 61, 71, 145, 155] [1, 6, 31, 42, 72, 79, 156, 163] [13, 21, 24, 32, 63, 73, 147, 157] [10, 20, 61, 71, 74, 81, 158, 165] [4, 11, 51, 62, 65, 75, 149, 159] [0, 3, 37, 48, 67, 76, 151, 160] [25, 35, 36, 46, 77, 82, 161, 166] [15, 26, 55, 66, 69, 78, 153, 162] [1, 6, 31, 42, 72, 79, 156, 163] [19, 30, 41, 52, 80, 83, 164, 167] [10, 20, 61, 71, 74, 81, 158, 165] [25, 35, 36, 46, 77, 82, 161, 166] [19, 30, 41, 52, 80, 83, 164, 167]

Code ID 168-84-2 · download JSON · raw on GitHub