[[192,24,6]] · QEC Challenge

[[192,24,6]] d ≤

192

kd²/n

4.5

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[6, 22, 23, 26, 54, 76]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[125, 154, 158, 162, 180, 183]

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 6

qubit degrees H_X 3 · H_Z 3

trapping sets H_X (1,3)×192 (2,4)×1440 (3,3)×96 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 192 (2,4): 1440 (3,3): 96 (3,5): 14112 (3,7): 1920

trapping sets H_Z (1,3)×192 (2,4)×1440 (3,3)×96 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 192 (2,4): 1440 (3,3): 96 (3,5): 14112 (3,7): 1920

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-18

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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[192,24,6]] — two-block group-algebra code on SmallGroup(96,14)

Direction & hypothesis

Target: unrestricted weight-6 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 = 24·36/192 = 4.50.

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_wt6wtL3_order96_k24.txt, line 2. This row is SmallGroup(96,14) with nonidentity GAP supports a=[2,32], b=[8,19]. 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=24), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 6, not an exact certificate. The published distance (d=6) 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

Union Alpha (maker currently anonymous), 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(96,14), 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=[2,32] and b=[8,19] 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 = 192 - rank(HX) - rank(HZ) = 24. 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 96 (max weight 6) · Z-checks 96 (max weight 6)

H_X (96 checks, sparse supports)

[0, 30, 32, 96, 119, 140] [0, 1, 72, 97, 154, 158] [2, 24, 53, 98, 126, 161] [3, 55, 57, 99, 103, 118] [1, 4, 59, 100, 166, 169] [5, 9, 11, 101, 145, 167] [6, 59, 61, 99, 102, 172] [2, 7, 67, 96, 103, 172] [3, 8, 88, 104, 129, 133] [4, 9, 22, 105, 131, 177] [5, 10, 46, 106, 130, 178] [6, 11, 19, 104, 107, 190] [12, 48, 75, 108, 136, 151] [13, 25, 77, 106, 109, 181] [14, 26, 51, 105, 110, 182] [15, 52, 79, 108, 111, 179] [8, 16, 80, 112, 142, 147] [17, 27, 29, 113, 121, 143] [18, 80, 82, 96, 114, 150] [10, 19, 31, 115, 144, 184] [11, 20, 83, 112, 116, 191] [21, 31, 33, 113, 117, 187] [10, 22, 83, 114, 118, 146] [12, 23, 85, 99, 119, 150] [13, 24, 68, 115, 120, 186] [14, 25, 87, 101, 121, 187] [13, 15, 26, 118, 119, 122] [16, 27, 44, 110, 123, 156] [17, 28, 70, 109, 124, 157] [18, 29, 41, 97, 125, 183] [19, 30, 47, 108, 126, 189] [0, 20, 31, 123, 127, 182] [4, 21, 32, 124, 128, 181] [22, 33, 45, 125, 129, 180] [34, 49, 89, 124, 130, 163] [35, 50, 73, 123, 131, 164] [36, 74, 91, 98, 132, 189] [7, 37, 52, 97, 133, 190] [38, 53, 92, 130, 134, 157] [39, 54, 77, 131, 135, 156] [25, 40, 78, 132, 136, 155] [28, 41, 56, 120, 137, 168] [29, 42, 93, 100, 138, 188] [43, 56, 58, 101, 139, 175] [28, 44, 93, 102, 122, 140] [32, 45, 60, 137, 141, 186] [30, 33, 46, 138, 142, 185] [1, 47, 60, 139, 143, 171] [34, 48, 86, 137, 144, 174] [35, 49, 94, 113, 145, 175] [34, 36, 50, 103, 140, 146] [37, 40, 51, 100, 147, 191] [14, 38, 52, 144, 148, 168] [37, 39, 53, 143, 145, 149] [38, 40, 54, 102, 146, 150] [41, 55, 71, 98, 151, 179] [3, 42, 56, 105, 152, 164] [16, 43, 57, 106, 153, 163] [44, 58, 69, 107, 154, 162] [5, 45, 59, 151, 155, 161] [6, 46, 60, 152, 156, 160] [20, 47, 61, 153, 157, 159] [23, 62, 74, 104, 158, 183] [63, 75, 95, 109, 159, 178] [64, 76, 89, 110, 160, 177] [49, 65, 90, 111, 161, 176] [26, 66, 78, 129, 158, 162] [51, 67, 79, 128, 159, 163] [7, 68, 92, 127, 160, 164] [57, 69, 81, 115, 165, 174] [55, 58, 70, 116, 166, 173] [8, 71, 81, 117, 149, 167] [9, 61, 72, 165, 168, 170] [62, 65, 73, 112, 169, 188] [35, 63, 74, 120, 170, 184] [62, 64, 75, 121, 167, 171] [63, 65, 76, 114, 122, 172] [2, 66, 77, 142, 169, 173] [39, 67, 78, 141, 170, 174] [66, 68, 79, 117, 171, 175] [17, 69, 80, 126, 136, 176] [18, 70, 81, 127, 135, 177] [42, 71, 82, 128, 134, 178] [21, 72, 83, 132, 176, 179] [50, 84, 90, 133, 154, 180] [73, 85, 91, 134, 153, 181] [23, 86, 95, 135, 152, 182] [24, 54, 87, 107, 180, 183] [27, 82, 88, 141, 148, 184] [12, 84, 89, 147, 166, 185] [64, 85, 90, 148, 165, 186] [84, 86, 91, 139, 149, 187] [15, 87, 92, 116, 185, 188] [43, 88, 93, 111, 155, 189] [48, 76, 94, 125, 162, 190] [36, 94, 95, 138, 173, 191]

H_Z (96 checks, sparse supports)

[0, 7, 18, 96, 97, 127] [1, 29, 37, 97, 100, 143] [2, 36, 55, 98, 103, 173] [3, 6, 23, 99, 104, 152] [4, 42, 51, 100, 105, 128] [5, 25, 43, 101, 106, 155] [6, 44, 54, 102, 107, 156] [3, 7, 50, 103, 133, 164] [8, 11, 62, 104, 112, 167] [9, 14, 56, 101, 105, 168] [10, 13, 57, 106, 115, 118] [11, 58, 87, 101, 107, 116] [12, 15, 30, 108, 119, 185] [13, 28, 63, 109, 120, 122] [14, 27, 64, 110, 121, 148] [15, 65, 93, 111, 122, 188] [16, 20, 73, 112, 123, 153] [17, 21, 49, 113, 124, 176] [18, 22, 76, 114, 125, 177] [19, 24, 69, 107, 115, 126] [20, 70, 92, 116, 127, 157] [21, 71, 79, 117, 128, 179] [3, 22, 26, 105, 118, 129] [0, 23, 26, 119, 158, 182] [24, 41, 74, 98, 120, 183] [17, 25, 75, 109, 121, 136] [26, 44, 76, 110, 122, 162] [27, 31, 35, 113, 123, 184] [28, 32, 34, 124, 137, 140] [29, 33, 94, 113, 125, 138] [2, 30, 80, 96, 126, 142] [31, 68, 81, 115, 117, 127] [32, 67, 82, 96, 128, 141] [8, 33, 66, 117, 129, 142] [10, 34, 38, 130, 144, 146] [9, 35, 39, 131, 145, 170] [36, 40, 83, 132, 146, 191] [8, 37, 84, 133, 147, 149] [38, 82, 85, 134, 148, 150] [39, 81, 86, 135, 149, 174] [12, 40, 80, 136, 147, 150] [41, 45, 48, 125, 137, 151] [42, 46, 95, 138, 152, 178] [43, 47, 91, 139, 153, 189] [0, 44, 50, 123, 140, 154] [45, 78, 88, 129, 141, 155] [16, 46, 77, 106, 142, 156] [17, 47, 53, 126, 143, 157] [19, 48, 52, 108, 144, 190] [5, 49, 53, 130, 145, 161] [22, 50, 54, 131, 146, 180] [16, 51, 89, 110, 147, 163] [52, 88, 90, 111, 133, 148] [53, 71, 91, 98, 134, 149] [18, 23, 54, 135, 150, 183] [12, 55, 59, 99, 151, 166] [56, 60, 86, 137, 139, 152] [57, 61, 85, 99, 153, 165] [1, 58, 84, 139, 154, 166] [40, 59, 93, 100, 102, 155] [27, 39, 60, 141, 143, 156] [28, 38, 61, 102, 157, 168] [1, 62, 66, 158, 169, 171] [61, 63, 67, 159, 170, 172] [60, 64, 68, 160, 171, 186] [2, 59, 65, 161, 169, 172] [58, 66, 94, 162, 173, 175] [34, 57, 67, 103, 163, 174] [35, 56, 68, 120, 164, 175] [69, 72, 90, 154, 165, 176] [4, 70, 89, 124, 166, 177] [5, 71, 75, 151, 167, 178] [41, 52, 72, 97, 168, 179] [4, 73, 77, 131, 169, 181] [72, 74, 78, 132, 158, 170] [47, 75, 79, 108, 159, 171] [6, 7, 76, 160, 172, 190] [70, 77, 95, 109, 135, 173] [48, 69, 78, 136, 162, 174] [43, 49, 79, 111, 163, 175] [65, 80, 83, 112, 114, 176] [9, 64, 81, 165, 167, 177] [10, 63, 82, 114, 178, 184] [15, 55, 83, 116, 118, 179] [33, 84, 87, 180, 185, 187] [13, 32, 85, 119, 181, 186] [14, 31, 86, 144, 182, 187] [29, 62, 87, 121, 183, 188] [19, 74, 88, 104, 184, 189] [46, 89, 92, 130, 160, 185] [24, 45, 90, 161, 180, 186] [21, 25, 91, 132, 181, 187] [42, 73, 92, 134, 164, 188] [30, 36, 93, 138, 140, 189] [11, 37, 94, 145, 190, 191] [20, 51, 95, 159, 182, 191]

Code ID 192-24-6 · download JSON · raw on GitHub