[[168,60,2]] · QEC Challenge

[[168,60,2]] d =

168

kd²/n

1.429

X/Z

Share this result

Distance

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

witness operator (support, 2 qubits)

[53, 64]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[139, 150]

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 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)×84 (2,0)×126 (3,3)×532 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 84 (1,4): 84 (2,0): 126 (2,4): 168 (2,5): 1008 (3,3): 532 (3,4): 1512 (3,5): 2016 (3,6): 4536 (3,7): 5040

trapping sets H_Z (1,3)×84 (2,0)×126 (3,3)×532 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 84 (1,4): 84 (2,0): 126 (2,4): 168 (2,5): 1008 (3,3): 532 (3,4): 1512 (3,5): 2016 (3,6): 4536 (3,7): 5040

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(84,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt7wtL3_order84_k60.txt, line 1. nonidentity supports a=[3, 9], b=[4, 8, 17].

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

[[168,60,2]] — two-block group-algebra code on SmallGroup(84,2)

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 = 1.43.

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_order84_k60.txt, line 1. This row is SmallGroup(84,2) with nonidentity GAP supports a=[3, 9], b=[4, 8, 17]. 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=60), 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 (d=2) 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(84,2), 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, 9], b=[4, 8, 17] 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) = 60. 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 7) · Z-checks 84 (max weight 7)

H_X (84 checks, sparse supports)

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

H_Z (84 checks, sparse supports)

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

Code ID 168-60-2 · download JSON · raw on GitHub