[[196,28,6]] · QEC Challenge

[[196,28,6]] d ≤

196

kd²/n

5.143

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)

[13, 60, 120, 131, 152, 170]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[4, 22, 33, 46, 67, 152]

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)×98 (2,2)×98 (3,2)×196 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 98 (1,4): 98 (2,2): 98 (2,4): 784 (2,6): 588 (3,2): 196 (3,4): 3332 (3,6): 8232 (3,8): 4410 (3,10): 392

trapping sets H_Z (1,2)×98 (2,2)×98 (3,2)×196 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 98 (1,4): 98 (2,2): 98 (2,4): 784 (2,6): 588 (3,2): 196 (3,4): 3332 (3,6): 8232 (3,8): 4410 (3,10): 392

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

[[196,28,6]] — two-block group-algebra code on SmallGroup(98,3)

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 = 5.14.

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_order98_k28.txt, line 4. This row is SmallGroup(98,3) with nonidentity GAP supports a=[4], b=[5, 9, 29]. 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=28), 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(98,3), 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=[4] and b=[5, 9, 29] 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 = 196 - rank(HX) - rank(HZ) = 28. 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 98 (max weight 6) · Z-checks 98 (max weight 6)

H_X (98 checks, sparse supports)

[0, 48, 98, 133, 147, 180] [1, 5, 99, 139, 140, 176] [2, 61, 99, 100, 117, 145] [0, 3, 101, 146, 171, 195] [4, 10, 98, 102, 113, 152] [5, 11, 103, 153, 166, 194] [6, 72, 102, 104, 126, 158] [2, 7, 105, 109, 153, 159] [3, 8, 98, 106, 160, 192] [9, 17, 100, 107, 121, 164] [10, 18, 106, 108, 146, 165] [11, 19, 99, 109, 154, 190] [12, 81, 107, 110, 137, 169] [6, 13, 111, 116, 165, 170] [7, 14, 100, 103, 112, 139] [8, 15, 101, 113, 147, 187] [16, 26, 104, 114, 131, 174] [17, 27, 112, 115, 159, 175] [18, 28, 101, 102, 116, 133] [19, 29, 103, 117, 140, 184] [20, 88, 114, 118, 150, 178] [12, 21, 119, 125, 175, 179] [13, 22, 104, 108, 120, 152] [14, 23, 99, 105, 121, 127] [15, 24, 106, 122, 180, 195] [25, 37, 110, 123, 143, 182] [26, 38, 120, 124, 170, 183] [27, 39, 105, 107, 125, 145] [28, 40, 98, 108, 122, 126] [29, 41, 109, 127, 176, 194] [30, 93, 123, 128, 162, 185] [20, 31, 129, 136, 183, 186] [21, 32, 110, 115, 130, 164] [22, 33, 102, 111, 131, 138] [23, 34, 112, 117, 132, 153] [24, 35, 113, 133, 171, 192] [36, 50, 118, 134, 156, 188] [37, 51, 130, 135, 179, 189] [38, 52, 111, 114, 136, 158] [39, 53, 100, 115, 132, 137] [40, 54, 113, 116, 138, 146] [41, 55, 117, 139, 166, 190] [42, 96, 134, 140, 172, 190] [30, 43, 141, 149, 189, 191] [31, 44, 118, 124, 142, 174] [32, 45, 107, 119, 143, 151] [33, 46, 120, 126, 144, 165] [34, 47, 109, 121, 139, 145] [35, 48, 122, 146, 160, 187] [49, 62, 128, 147, 167, 192] [50, 63, 142, 148, 186, 193] [51, 64, 119, 123, 149, 169] [52, 65, 104, 124, 144, 150] [53, 66, 121, 125, 151, 159] [54, 67, 106, 126, 133, 152] [1, 55, 127, 153, 154, 184] [42, 56, 154, 161, 193, 194] [43, 57, 128, 135, 155, 182] [44, 58, 114, 129, 156, 163] [45, 59, 130, 137, 157, 175] [46, 60, 116, 131, 152, 158] [47, 61, 103, 127, 132, 159] [62, 73, 155, 160, 191, 195] [63, 74, 129, 134, 161, 178] [64, 75, 110, 135, 157, 162] [65, 76, 131, 136, 163, 170] [66, 77, 112, 137, 145, 164] [4, 67, 101, 122, 138, 165] [56, 68, 140, 148, 166, 188] [57, 69, 123, 141, 167, 173] [58, 70, 142, 150, 168, 183] [59, 71, 125, 143, 164, 169] [60, 72, 108, 138, 144, 170] [73, 82, 141, 147, 171, 185] [74, 83, 118, 148, 168, 172] [75, 84, 143, 149, 173, 179] [76, 85, 120, 150, 158, 174] [9, 77, 105, 132, 151, 175] [68, 78, 134, 154, 176, 181] [69, 79, 155, 162, 177, 189] [70, 80, 136, 156, 174, 178] [71, 81, 115, 151, 157, 179] [82, 89, 128, 160, 177, 180] [83, 90, 156, 161, 181, 186] [84, 91, 130, 162, 169, 182] [16, 85, 111, 144, 163, 183] [78, 86, 166, 172, 184, 193] [79, 87, 149, 167, 182, 185] [80, 88, 124, 163, 168, 186] [89, 94, 167, 171, 187, 191] [90, 95, 142, 172, 178, 188] [25, 91, 119, 157, 173, 189] [86, 92, 161, 176, 188, 190] [87, 93, 135, 173, 177, 191] [94, 97, 155, 180, 185, 192] [36, 95, 129, 168, 181, 193] [92, 96, 148, 181, 184, 194] [49, 97, 141, 177, 187, 195]

H_Z (98 checks, sparse supports)

[0, 4, 8, 28, 98, 101] [1, 2, 11, 23, 99, 153] [2, 9, 14, 39, 100, 105] [3, 15, 18, 67, 101, 106] [4, 6, 18, 33, 102, 165] [5, 14, 19, 61, 99, 103] [6, 16, 22, 52, 104, 111] [7, 23, 27, 77, 105, 112] [8, 10, 24, 54, 106, 113] [9, 12, 27, 45, 107, 175] [10, 22, 28, 72, 102, 108] [7, 11, 29, 47, 103, 109] [12, 25, 32, 64, 110, 119] [13, 33, 38, 85, 111, 120] [14, 17, 34, 66, 112, 121] [4, 15, 35, 40, 113, 122] [16, 20, 38, 58, 114, 183] [17, 32, 39, 81, 107, 115] [13, 18, 40, 60, 108, 116] [2, 19, 34, 41, 109, 117] [20, 36, 44, 74, 118, 129] [21, 45, 51, 91, 119, 130] [22, 26, 46, 76, 120, 131] [9, 23, 47, 53, 121, 132] [24, 28, 48, 67, 122, 133] [25, 30, 51, 69, 123, 189] [26, 44, 52, 88, 114, 124] [21, 27, 53, 71, 115, 125] [6, 28, 46, 54, 116, 126] [23, 29, 55, 61, 117, 127] [30, 49, 57, 82, 128, 141] [31, 58, 63, 95, 129, 142] [32, 37, 59, 84, 130, 143] [16, 33, 60, 65, 131, 144] [34, 39, 61, 77, 132, 145] [0, 18, 35, 54, 133, 146] [36, 42, 63, 78, 134, 193] [37, 57, 64, 93, 123, 135] [31, 38, 65, 80, 124, 136] [12, 39, 59, 66, 125, 137] [33, 40, 67, 72, 126, 138] [1, 14, 41, 47, 127, 139] [1, 19, 42, 68, 140, 154] [43, 69, 73, 97, 141, 155] [44, 50, 70, 90, 142, 156] [25, 45, 71, 75, 143, 157] [46, 52, 72, 85, 144, 158] [2, 27, 47, 66, 145, 159] [3, 10, 40, 48, 98, 146] [0, 15, 49, 73, 147, 195] [50, 68, 74, 96, 134, 148] [43, 51, 75, 87, 135, 149] [20, 52, 70, 76, 136, 150] [45, 53, 77, 81, 137, 151] [4, 22, 54, 60, 138, 152] [5, 7, 34, 55, 139, 153] [11, 55, 56, 78, 154, 166] [57, 62, 79, 94, 155, 167] [36, 58, 80, 83, 156, 168] [59, 64, 81, 91, 157, 169] [6, 38, 60, 76, 158, 170] [7, 17, 53, 61, 100, 159] [8, 48, 62, 82, 147, 160] [56, 63, 83, 92, 148, 161] [30, 64, 79, 84, 149, 162] [58, 65, 85, 88, 150, 163] [9, 32, 66, 71, 151, 164] [10, 13, 46, 67, 152, 165] [5, 41, 68, 86, 166, 176] [49, 69, 87, 89, 167, 177] [70, 74, 88, 95, 168, 178] [12, 51, 71, 84, 169, 179] [13, 26, 65, 72, 104, 170] [3, 35, 73, 89, 160, 171] [42, 74, 86, 90, 161, 172] [69, 75, 91, 93, 162, 173] [16, 44, 76, 80, 163, 174] [17, 21, 59, 77, 164, 175] [1, 29, 78, 92, 176, 184] [79, 82, 93, 97, 177, 185] [20, 63, 80, 90, 178, 186] [21, 37, 75, 81, 110, 179] [0, 24, 82, 94, 171, 180] [78, 83, 95, 96, 172, 181] [25, 57, 84, 87, 173, 182] [26, 31, 70, 85, 174, 183] [19, 55, 86, 96, 184, 190] [30, 73, 87, 94, 185, 191] [31, 50, 83, 88, 118, 186] [15, 48, 89, 97, 180, 187] [36, 68, 90, 92, 181, 188] [37, 43, 79, 91, 182, 189] [11, 41, 42, 92, 190, 194] [43, 62, 89, 93, 128, 191] [8, 35, 49, 94, 187, 192] [50, 56, 86, 95, 188, 193] [5, 29, 56, 96, 140, 194] [3, 24, 62, 97, 192, 195]

Code ID 196-28-6 · download JSON · raw on GitHub