[[192,20,8]] · QEC Challenge

[[192,20,8]] d ≤

192

kd²/n

6.667

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · 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 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[10, 21, 25, 39, 106, 121, 142, 163]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[10, 21, 25, 39, 106, 121, 142, 163]

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 8 · H_Z 8 (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 (3,5)×14400 (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 (3,5): 14400 (3,7): 1920

trapping sets H_Z (1,3)×192 (2,4)×1440 (3,5)×14400 (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 (3,5): 14400 (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

[[192,20,8]] — two-block group-algebra code on SmallGroup(96,9)

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 = 20·64/192 = 6.67.

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_k20.txt, line 2. This row is SmallGroup(96,9) with nonidentity GAP supports a=[2,30], b=[8,70]. 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=20), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 8, not an exact certificate. The published distance (d=8) 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(96,9), 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=[2,30] and b=[8,70] 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) = 20. 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, 46, 55, 96, 162, 167] [1, 17, 59, 97, 173, 178] [2, 67, 73, 98, 141, 182] [3, 30, 88, 99, 118, 180] [4, 8, 47, 100, 111, 166] [5, 27, 72, 101, 134, 140] [4, 6, 80, 102, 113, 183] [7, 35, 77, 103, 155, 187] [0, 8, 56, 104, 129, 185] [9, 11, 41, 105, 122, 177] [3, 10, 31, 106, 148, 154] [11, 43, 83, 107, 124, 188] [12, 51, 94, 108, 136, 165] [13, 23, 68, 102, 109, 181] [14, 48, 87, 110, 116, 161] [13, 15, 89, 111, 131, 168] [1, 16, 44, 112, 132, 168] [9, 17, 70, 113, 143, 159] [18, 41, 59, 96, 114, 190] [19, 22, 28, 115, 135, 184] [5, 20, 29, 112, 116, 136] [19, 21, 56, 99, 117, 163] [20, 22, 93, 118, 133, 139] [2, 23, 74, 119, 150, 176] [24, 26, 62, 107, 120, 186] [12, 25, 52, 121, 127, 172] [26, 64, 92, 122, 145, 179] [4, 27, 57, 123, 146, 179] [5, 28, 80, 124, 157, 170] [6, 29, 81, 97, 125, 191] [16, 30, 32, 126, 149, 189] [31, 33, 69, 123, 127, 150] [18, 32, 60, 104, 128, 174] [30, 33, 71, 129, 147, 153] [7, 34, 65, 114, 130, 183] [24, 35, 85, 131, 138, 164] [36, 62, 77, 98, 132, 184] [37, 40, 49, 117, 133, 190] [14, 38, 50, 118, 130, 134] [37, 39, 74, 108, 135, 142] [38, 40, 95, 115, 136, 160] [10, 41, 71, 137, 142, 160] [3, 11, 42, 115, 138, 161] [16, 31, 43, 101, 139, 181] [44, 69, 83, 102, 140, 158] [0, 45, 57, 137, 141, 164] [21, 46, 58, 98, 138, 142] [45, 47, 81, 109, 114, 143] [13, 48, 75, 125, 144, 188] [14, 49, 89, 145, 152, 175] [15, 50, 90, 103, 146, 189] [34, 51, 53, 128, 147, 191] [52, 54, 84, 129, 144, 148] [36, 53, 78, 119, 149, 156] [51, 54, 86, 126, 150, 171] [19, 29, 55, 151, 156, 171] [20, 56, 82, 126, 152, 172] [21, 57, 93, 106, 153, 186] [22, 27, 58, 107, 154, 169] [42, 59, 61, 151, 155, 175] [1, 60, 88, 103, 152, 156] [9, 44, 61, 120, 125, 157] [25, 62, 86, 139, 158, 163] [12, 26, 63, 133, 140, 159] [34, 52, 64, 110, 160, 166] [65, 84, 92, 111, 137, 161] [2, 66, 75, 143, 158, 162] [39, 67, 76, 96, 159, 163] [66, 68, 90, 100, 132, 164] [17, 32, 69, 100, 165, 182] [18, 33, 70, 108, 141, 166] [42, 60, 71, 117, 130, 167] [6, 72, 82, 110, 165, 168] [37, 50, 73, 153, 169, 174] [38, 74, 91, 147, 154, 170] [39, 75, 95, 121, 171, 177] [40, 48, 76, 122, 151, 172] [63, 77, 79, 157, 169, 173] [7, 78, 94, 97, 170, 174] [24, 65, 79, 105, 146, 175] [45, 58, 80, 105, 176, 187] [28, 46, 81, 119, 155, 177] [47, 55, 82, 128, 144, 178] [10, 70, 83, 121, 176, 179] [35, 53, 84, 109, 167, 180] [36, 54, 85, 99, 162, 181] [63, 78, 86, 112, 135, 182] [15, 87, 91, 101, 180, 183] [43, 61, 88, 116, 131, 184] [66, 76, 89, 120, 178, 185] [49, 67, 90, 104, 173, 186] [68, 73, 91, 123, 149, 187] [25, 85, 92, 106, 185, 188] [8, 72, 93, 127, 145, 189] [64, 79, 94, 113, 134, 190] [23, 87, 95, 124, 148, 191]

H_Z (96 checks, sparse supports)

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

Code ID 192-20-8 · download JSON · raw on GitHub