[[200,2,20]] · QEC Challenge

[[200,2,20]] d ≤

200

kd²/n

4.0

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)

[10, 23, 40, 51, 74, 80, 84, 99, 122, 124, 132, 138, 151, 169, 172, 176, 179, 180, 186, 198]

d_Z 20 · witness weight 20 (claimed upper_bound)

witness operator (support, 20 qubits)

[15, 24, 27, 36, 53, 56, 57, 67, 83, 89, 90, 92, 125, 126, 127, 137, 143, 144, 162, 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)×100 (2,2)×100 (3,2)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 100 (1,4): 100 (2,2): 100 (2,4): 800 (2,6): 600 (3,2): 100 (3,4): 3200 (3,6): 9600 (3,8): 5400 (3,10): 400

trapping sets H_Z (1,2)×100 (2,2)×100 (3,2)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 100 (1,4): 100 (2,2): 100 (2,4): 800 (2,6): 600 (3,2): 100 (3,4): 3200 (3,6): 9600 (3,8): 5400 (3,10): 400

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

[[200,2,20]] — two-block group-algebra code on SmallGroup(100,1)

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 = 4.00.

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_order100_k2.txt, line 7. This row is SmallGroup(100,1) with nonidentity GAP supports a=[9], b=[2, 5, 40]. 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(100,1), 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=[9] and b=[2, 5, 40] 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 = 200 - 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 100 (max weight 6) · Z-checks 100 (max weight 6)

H_X (100 checks, sparse supports)

[0, 68, 100, 105, 140, 185] [1, 13, 100, 101, 149, 190] [2, 55, 101, 102, 154, 192] [2, 3, 103, 112, 158, 175] [4, 82, 100, 104, 163, 194] [5, 6, 102, 105, 163, 195] [6, 25, 106, 117, 155, 167] [7, 26, 101, 107, 140, 197] [0, 8, 108, 120, 164, 171] [9, 72, 102, 109, 149, 198] [8, 10, 110, 123, 160, 174] [9, 11, 103, 111, 124, 159] [12, 92, 104, 112, 136, 144] [13, 15, 113, 127, 168, 178] [14, 16, 105, 114, 154, 199] [15, 41, 115, 130, 137, 181] [16, 42, 106, 116, 131, 136] [17, 43, 107, 117, 124, 145] [3, 18, 118, 134, 146, 184] [4, 19, 108, 119, 135, 145] [20, 85, 109, 120, 131, 150] [18, 21, 121, 138, 142, 186] [19, 22, 110, 122, 139, 141] [20, 23, 111, 123, 140, 199] [24, 98, 112, 124, 127, 155] [25, 28, 125, 143, 151, 189] [26, 29, 113, 126, 144, 150] [27, 30, 114, 127, 135, 159] [28, 59, 122, 128, 147, 191] [29, 60, 115, 121, 129, 148] [30, 61, 116, 130, 149, 194] [31, 62, 112, 117, 131, 164] [10, 32, 129, 132, 152, 193] [11, 33, 118, 128, 133, 153] [12, 34, 119, 134, 154, 197] [35, 94, 117, 120, 135, 168] [32, 36, 126, 136, 156, 194] [33, 37, 121, 125, 137, 157] [34, 38, 122, 138, 158, 196] [35, 39, 100, 123, 139, 195] [40, 50, 114, 124, 140, 172] [41, 45, 133, 141, 161, 196] [42, 46, 125, 132, 142, 162] [43, 47, 126, 143, 163, 198] [44, 48, 120, 127, 144, 175] [45, 63, 111, 145, 165, 197] [46, 75, 110, 128, 146, 166] [47, 76, 129, 147, 167, 186] [48, 77, 101, 130, 148, 185] [49, 78, 104, 131, 149, 179] [21, 50, 116, 150, 169, 198] [22, 51, 115, 132, 151, 170] [23, 52, 133, 152, 171, 191] [24, 53, 102, 134, 153, 190] [36, 54, 107, 135, 154, 182] [51, 55, 113, 136, 155, 173] [52, 56, 137, 156, 174, 189] [53, 57, 103, 138, 157, 188] [54, 58, 104, 139, 158, 187] [49, 59, 119, 159, 176, 199] [60, 64, 118, 141, 160, 177] [61, 65, 142, 161, 178, 193] [62, 66, 105, 143, 162, 192] [63, 67, 109, 144, 163, 187] [5, 64, 103, 145, 164, 180] [65, 87, 146, 165, 174, 181] [66, 88, 106, 147, 166, 173] [67, 89, 107, 148, 167, 172] [37, 68, 106, 150, 168, 183] [38, 69, 151, 169, 181, 184] [39, 70, 108, 152, 170, 180] [40, 71, 109, 153, 171, 179] [69, 72, 155, 172, 178, 186] [70, 73, 110, 156, 173, 177] [71, 74, 111, 157, 174, 176] [1, 75, 108, 159, 175, 188] [76, 79, 160, 176, 184, 189] [77, 80, 113, 161, 177, 183] [78, 81, 114, 162, 178, 182] [14, 79, 158, 164, 179, 191] [80, 95, 115, 157, 165, 180] [81, 96, 116, 156, 166, 181] [56, 82, 167, 168, 182, 193] [57, 83, 118, 166, 169, 183] [58, 84, 119, 165, 170, 184] [83, 85, 121, 162, 172, 185] [84, 86, 122, 161, 173, 186] [7, 87, 171, 175, 187, 196] [88, 90, 125, 170, 176, 188] [89, 91, 126, 169, 177, 189] [27, 90, 128, 139, 179, 190] [91, 99, 129, 138, 180, 191] [73, 92, 132, 148, 182, 192] [74, 93, 133, 147, 183, 193] [93, 94, 137, 143, 185, 194] [17, 95, 141, 153, 187, 195] [96, 97, 142, 152, 188, 196] [44, 97, 123, 146, 190, 197] [86, 98, 130, 151, 192, 198] [31, 99, 134, 160, 195, 199]

H_Z (100 checks, sparse supports)

[0, 1, 4, 39, 100, 108] [1, 2, 7, 48, 101, 175] [2, 5, 9, 53, 102, 103] [3, 11, 57, 64, 103, 118] [4, 12, 49, 58, 104, 119] [0, 5, 14, 62, 105, 164] [6, 16, 66, 68, 105, 106] [7, 17, 54, 67, 107, 187] [8, 19, 70, 75, 108, 110] [9, 20, 63, 71, 109, 111] [10, 22, 46, 73, 110, 132] [11, 23, 45, 74, 111, 133] [3, 12, 24, 31, 112, 134] [13, 26, 55, 77, 101, 113] [14, 27, 40, 78, 114, 179] [15, 29, 51, 80, 113, 115] [16, 30, 50, 81, 114, 116] [6, 17, 31, 35, 117, 195] [18, 33, 60, 83, 118, 121] [19, 34, 59, 84, 119, 122] [8, 20, 35, 44, 120, 123] [21, 29, 37, 85, 121, 150] [22, 28, 38, 86, 122, 151] [10, 23, 39, 97, 123, 152] [11, 17, 24, 40, 124, 153] [25, 37, 42, 88, 106, 125] [26, 36, 43, 89, 107, 126] [13, 24, 27, 44, 127, 190] [28, 33, 46, 90, 125, 128] [29, 32, 47, 91, 126, 129] [15, 30, 48, 98, 127, 130] [16, 20, 31, 49, 131, 199] [32, 42, 51, 92, 132, 136] [33, 41, 52, 93, 133, 137] [18, 34, 53, 99, 134, 138] [19, 27, 35, 54, 135, 139] [12, 16, 36, 55, 136, 154] [15, 37, 56, 94, 137, 168] [21, 38, 57, 91, 138, 169] [22, 39, 58, 90, 139, 170] [0, 7, 23, 40, 140, 171] [22, 41, 60, 95, 115, 141] [21, 42, 61, 96, 116, 142] [25, 43, 62, 94, 117, 143] [12, 26, 44, 63, 144, 197] [17, 19, 45, 64, 141, 145] [18, 46, 65, 97, 142, 146] [28, 47, 66, 93, 143, 147] [29, 48, 67, 92, 144, 148] [1, 9, 30, 49, 149, 159] [20, 26, 50, 68, 140, 150] [25, 51, 69, 98, 151, 155] [32, 52, 70, 96, 152, 156] [33, 53, 71, 95, 153, 157] [2, 14, 34, 54, 154, 158] [6, 24, 55, 72, 102, 155] [36, 56, 73, 81, 156, 182] [37, 57, 74, 80, 157, 183] [3, 38, 58, 79, 158, 184] [11, 27, 59, 75, 128, 159] [10, 60, 76, 99, 129, 160] [41, 61, 77, 86, 130, 161] [42, 62, 78, 85, 131, 162] [4, 5, 43, 63, 145, 163] [8, 31, 64, 79, 160, 164] [45, 65, 80, 84, 161, 165] [46, 66, 81, 83, 162, 166] [6, 47, 67, 82, 163, 167] [13, 35, 68, 82, 100, 168] [50, 69, 83, 89, 169, 172] [51, 70, 84, 88, 170, 173] [8, 52, 71, 87, 171, 174] [40, 67, 72, 85, 109, 172] [55, 66, 73, 86, 173, 192] [10, 56, 65, 74, 174, 193] [3, 44, 75, 87, 146, 175] [59, 74, 76, 88, 147, 176] [60, 73, 77, 89, 148, 177] [13, 61, 72, 78, 149, 178] [49, 71, 79, 90, 176, 179] [64, 70, 80, 91, 177, 180] [15, 65, 69, 81, 178, 181] [54, 78, 82, 92, 104, 182] [68, 77, 83, 93, 183, 185] [18, 69, 76, 84, 184, 186] [0, 48, 85, 94, 120, 185] [21, 47, 72, 86, 186, 198] [58, 63, 87, 95, 165, 187] [57, 75, 88, 96, 166, 188] [25, 56, 76, 89, 167, 189] [1, 53, 90, 97, 188, 190] [28, 52, 79, 91, 189, 191] [2, 62, 92, 98, 112, 192] [32, 61, 82, 93, 193, 194] [4, 30, 36, 94, 135, 194] [5, 39, 95, 99, 180, 195] [38, 41, 87, 96, 181, 196] [7, 34, 45, 97, 196, 197] [9, 43, 50, 98, 124, 198] [14, 23, 59, 99, 191, 199]

Code ID 200-2-20 · download JSON · raw on GitHub