[[168,48,3]] · QEC Challenge

[[168,48,3]] d =

168

kd²/n

2.571

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[38, 62, 73]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[23, 33, 43]

certificate exact, d = 3 · scipy/HiGHS MILP

X: no logical < 3 exists; Z: no logical < 3 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)×168 (2,4)×420 (3,0)×112 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 168 (2,4): 420 (2,6): 1512 (3,0): 112 (3,4): 588 (3,6): 7392 (3,8): 23856 (3,10): 672

trapping sets H_Z (1,4)×168 (2,4)×420 (3,0)×112 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 168 (2,4): 420 (2,6): 1512 (3,0): 112 (3,4): 588 (3,6): 7392 (3,8): 23856 (3,10): 672

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_wt8wtL4_order84_k48.txt, line 1. nonidentity supports a=[2, 3, 6], b=[5, 13, 36].

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

[[168,48,3]] — 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 = 2.57.

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_order84_k48.txt, line 1. This row is SmallGroup(84,1) with nonidentity GAP supports a=[2, 3, 6], b=[5, 13, 36]. 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=48), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 3, 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=[2, 3, 6], b=[5, 13, 36] 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) = 48. 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, 6, 8, 24, 84, 107, 131, 143] [0, 1, 8, 13, 85, 113, 137, 149] [0, 2, 6, 14, 86, 117, 141, 153] [1, 3, 13, 18, 87, 118, 142, 154] [4, 16, 19, 36, 84, 88, 119, 143] [0, 1, 2, 5, 89, 123, 147, 158] [3, 6, 18, 24, 90, 124, 148, 159] [4, 7, 19, 25, 85, 91, 125, 149] [2, 8, 14, 24, 92, 127, 151, 161] [1, 3, 5, 9, 93, 128, 152, 162] [4, 10, 16, 26, 86, 94, 129, 153] [7, 11, 25, 30, 87, 95, 130, 154] [12, 28, 31, 48, 84, 88, 96, 131] [2, 5, 8, 13, 97, 133, 156, 164] [3, 6, 9, 14, 98, 134, 157, 165] [4, 7, 10, 15, 89, 99, 135, 158] [11, 16, 30, 36, 90, 100, 136, 159] [12, 17, 31, 37, 85, 91, 101, 137] [5, 9, 13, 18, 102, 138, 160, 166] [10, 19, 26, 36, 92, 103, 139, 161] [7, 11, 15, 20, 93, 104, 140, 162] [12, 21, 28, 38, 86, 94, 105, 141] [17, 22, 37, 42, 87, 95, 106, 142] [23, 40, 43, 60, 88, 96, 107, 143] [9, 14, 18, 24, 108, 144, 163, 167] [10, 15, 19, 25, 97, 109, 145, 164] [11, 16, 20, 26, 98, 110, 146, 165] [12, 17, 21, 27, 89, 99, 111, 147] [22, 28, 42, 48, 90, 100, 112, 148] [23, 29, 43, 49, 91, 101, 113, 149] [15, 20, 25, 30, 102, 114, 150, 166] [21, 31, 38, 48, 92, 103, 115, 151] [17, 22, 27, 32, 93, 104, 116, 152] [23, 33, 40, 50, 94, 105, 117, 153] [29, 34, 49, 54, 95, 106, 118, 154] [35, 52, 55, 71, 84, 96, 107, 119] [20, 26, 30, 36, 108, 120, 155, 167] [21, 27, 31, 37, 97, 109, 121, 156] [22, 28, 32, 38, 98, 110, 122, 157] [23, 29, 33, 39, 99, 111, 123, 158] [34, 40, 54, 60, 100, 112, 124, 159] [35, 41, 55, 61, 85, 101, 113, 125] [27, 32, 37, 42, 102, 114, 126, 160] [33, 43, 50, 60, 103, 115, 127, 161] [29, 34, 39, 44, 104, 116, 128, 162] [35, 45, 52, 62, 86, 105, 117, 129] [41, 46, 61, 66, 87, 106, 118, 130] [47, 64, 67, 79, 88, 107, 119, 131] [32, 38, 42, 48, 108, 120, 132, 163] [33, 39, 43, 49, 109, 121, 133, 164] [34, 40, 44, 50, 110, 122, 134, 165] [35, 41, 45, 51, 89, 111, 123, 135] [46, 52, 66, 71, 90, 112, 124, 136] [47, 53, 67, 72, 91, 113, 125, 137] [39, 44, 49, 54, 114, 126, 138, 166] [45, 55, 62, 71, 92, 115, 127, 139] [41, 46, 51, 56, 93, 116, 128, 140] [47, 57, 64, 73, 94, 117, 129, 141] [53, 58, 72, 76, 95, 118, 130, 142] [59, 75, 77, 83, 96, 119, 131, 143] [44, 50, 54, 60, 120, 132, 144, 167] [45, 51, 55, 61, 97, 121, 133, 145] [46, 52, 56, 62, 98, 122, 134, 146] [47, 53, 57, 63, 99, 123, 135, 147] [58, 64, 76, 79, 100, 124, 136, 148] [59, 65, 77, 80, 101, 125, 137, 149] [51, 56, 61, 66, 102, 126, 138, 150] [57, 67, 73, 79, 103, 127, 139, 151] [53, 58, 63, 68, 104, 128, 140, 152] [59, 69, 75, 81, 105, 129, 141, 153] [65, 70, 80, 82, 106, 130, 142, 154] [56, 62, 66, 71, 108, 132, 144, 155] [57, 63, 67, 72, 109, 133, 145, 156] [58, 64, 68, 73, 110, 134, 146, 157] [59, 65, 69, 74, 111, 135, 147, 158] [70, 75, 82, 83, 112, 136, 148, 159] [63, 68, 72, 76, 114, 138, 150, 160] [69, 77, 81, 83, 115, 139, 151, 161] [65, 70, 74, 78, 116, 140, 152, 162] [68, 73, 76, 79, 120, 144, 155, 163] [69, 74, 77, 80, 121, 145, 156, 164] [70, 75, 78, 81, 122, 146, 157, 165] [74, 78, 80, 82, 126, 150, 160, 166] [78, 81, 82, 83, 132, 155, 163, 167]

H_Z (84 checks, sparse supports)

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

Code ID 168-48-3 · download JSON · raw on GitHub