[[192,2,20]] · QEC Challenge

[[192,2,20]] d ≤

192

kd²/n

4.167

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 20, d_Z ≤ 20 · 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 20 · witness weight 20 (claimed upper_bound)

witness operator (support, 20 qubits)

[11, 12, 13, 18, 34, 51, 67, 70, 81, 94, 105, 107, 131, 137, 139, 164, 167, 175, 178, 179]

d_Z 20 · witness weight 20 (claimed upper_bound)

witness operator (support, 20 qubits)

[2, 23, 36, 37, 48, 56, 60, 65, 74, 89, 90, 91, 110, 134, 135, 139, 145, 154, 159, 182]

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 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

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

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

(1,2): 96 (1,4): 96 (2,2): 96 (2,4): 768 (2,6): 576 (3,2): 96 (3,4): 3168 (3,6): 8736 (3,8): 5760 (3,10): 384

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

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

(1,2): 96 (1,4): 96 (2,2): 96 (2,4): 768 (2,6): 576 (3,2): 96 (3,4): 3168 (3,6): 8736 (3,8): 5760 (3,10): 384

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

[[192,2,20]] — two-block group-algebra code on SmallGroup(96,48)

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 = 4.17.

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_wt6wtL2_order96_k2.txt, line 155. This row is SmallGroup(96,48) with nonidentity GAP supports a=[9], b=[15, 18, 57]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 20, not an exact certificate. The published distance (d=20) 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(96,48), 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=[9] and b=[15, 18, 57] 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) = 2. 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, 82, 96, 126, 136, 137] [1, 16, 97, 142, 150, 151] [2, 91, 98, 118, 158, 173] [3, 33, 99, 100, 153, 163] [4, 93, 100, 112, 155, 164] [5, 27, 98, 101, 156, 165] [6, 56, 102, 104, 110, 166] [7, 34, 103, 129, 169, 183] [0, 8, 104, 105, 167, 174] [9, 41, 105, 123, 168, 175] [10, 42, 99, 103, 106, 176] [11, 43, 107, 114, 121, 177] [12, 54, 108, 109, 142, 186] [13, 95, 109, 130, 143, 147] [14, 48, 96, 110, 180, 188] [15, 74, 101, 111, 145, 181] [16, 60, 107, 112, 113, 182] [17, 61, 96, 113, 177, 183] [1, 18, 108, 114, 116, 178] [10, 19, 115, 117, 123, 132] [20, 55, 109, 116, 138, 179] [21, 80, 117, 124, 134, 139] [22, 57, 111, 118, 125, 184] [2, 23, 119, 120, 156, 190] [24, 62, 120, 144, 157, 163] [25, 63, 97, 121, 131, 185] [26, 64, 106, 122, 161, 186] [3, 27, 118, 123, 124, 187] [4, 28, 97, 124, 184, 188] [5, 29, 112, 119, 125, 127] [6, 30, 126, 128, 138, 146] [31, 69, 113, 120, 127, 152] [32, 70, 128, 140, 148, 153] [33, 71, 115, 122, 129, 189] [34, 78, 130, 131, 149, 167] [35, 79, 98, 131, 168, 171] [7, 36, 99, 132, 134, 191] [25, 37, 114, 133, 135, 169] [38, 73, 100, 134, 159, 174] [39, 89, 116, 119, 135, 160] [40, 75, 102, 136, 172, 190] [41, 83, 99, 128, 137, 190] [8, 42, 129, 130, 138, 140] [29, 43, 97, 139, 141, 159] [9, 44, 101, 131, 140, 189] [31, 45, 102, 141, 151, 161] [11, 46, 133, 142, 143, 152] [47, 81, 135, 143, 154, 167] [12, 48, 144, 145, 164, 178] [13, 49, 103, 145, 179, 182] [14, 50, 104, 146, 148, 158] [15, 51, 125, 147, 149, 180] [52, 84, 105, 108, 148, 170] [53, 85, 127, 132, 149, 171] [54, 86, 107, 150, 162, 191] [17, 55, 104, 143, 151, 191] [18, 56, 96, 144, 152, 154] [19, 57, 101, 153, 155, 170] [20, 58, 106, 137, 145, 154] [21, 59, 107, 155, 165, 172] [22, 60, 139, 147, 156, 157] [61, 88, 141, 149, 157, 178] [62, 92, 108, 122, 158, 184] [23, 63, 112, 159, 161, 175] [50, 64, 120, 138, 160, 162] [24, 65, 110, 113, 161, 187] [52, 66, 111, 140, 144, 162] [26, 67, 115, 163, 164, 185] [68, 90, 117, 146, 164, 182] [28, 69, 114, 157, 158, 165] [58, 70, 105, 115, 166, 180] [30, 71, 106, 160, 167, 168] [32, 72, 118, 162, 168, 176] [35, 73, 119, 136, 169, 189] [36, 74, 109, 123, 170, 172] [37, 75, 134, 152, 171, 173] [38, 76, 121, 124, 130, 172] [39, 77, 122, 154, 159, 173] [40, 78, 126, 174, 175, 181] [79, 94, 128, 133, 175, 187] [44, 80, 100, 125, 169, 176] [45, 81, 116, 126, 177, 185] [46, 82, 102, 171, 178, 179] [47, 83, 129, 166, 173, 179] [49, 84, 132, 137, 150, 180] [76, 85, 103, 133, 165, 181] [51, 86, 139, 148, 182, 183] [53, 87, 136, 141, 170, 183] [59, 88, 127, 142, 181, 184] [65, 89, 98, 146, 151, 185] [66, 90, 110, 147, 176, 186] [67, 91, 135, 153, 187, 188] [68, 92, 150, 155, 160, 188] [72, 93, 117, 156, 186, 189] [77, 94, 121, 163, 166, 190] [87, 95, 111, 174, 177, 191]

H_Z (96 checks, sparse supports)

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

Code ID 192-2-20 · download JSON · raw on GitHub