[[200,60,2]] · QEC Challenge

[[200,60,2]] d =

200

kd²/n

1.2

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[115, 128]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[43, 61]

certificate exact, d = 2 · scipy/HiGHS MILP

X: no logical < 2 exists; Z: no logical < 2 exists

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 4 · H_Z 4 (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 (2,0)×100 (3,0)×80 (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 (2,0): 100 (2,2): 80 (2,4): 640 (2,6): 400 (3,0): 80 (3,2): 360 (3,4): 3000 (3,6): 5280 (3,8): 2880

trapping sets H_Z (1,2)×100 (2,0)×50 (3,2)×640 (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 (2,0): 50 (2,2): 80 (2,4): 940 (3,2): 640 (3,4): 4620 (3,6): 2280 (3,8): 200

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(100,9), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order100_k60.txt, line 1. nonidentity supports a=[4], b=[4, 16, 29].

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-19

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

[[200,60,2]] — two-block group-algebra code on SmallGroup(100,9)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.20.

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_k60.txt, line 1. This row is SmallGroup(100,9) with nonidentity GAP supports a=[4], b=[4, 16, 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=60), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, not an exact certificate. The published distance (d=2) 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

DeepSeek V4 Flash 0731 (provenance.model), 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(100,9), 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=[4], b=[4, 16, 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 = 200 - rank(HX) - rank(HZ) = 60. 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, 3, 100, 103, 187, 188] [1, 6, 101, 106, 169, 192] [2, 9, 102, 109, 130, 131] [0, 3, 100, 103, 176, 195] [4, 11, 104, 141, 158, 159] [5, 14, 105, 112, 114, 139] [1, 6, 101, 106, 182, 183] [7, 16, 107, 132, 150, 167] [8, 19, 108, 119, 146, 147] [2, 9, 102, 109, 117, 148] [10, 21, 101, 106, 110, 174] [4, 11, 111, 140, 141, 159] [12, 23, 112, 139, 176, 195] [13, 26, 113, 122, 126, 156] [5, 14, 105, 114, 123, 124] [15, 28, 100, 103, 115, 181] [7, 16, 116, 132, 149, 150] [17, 30, 117, 148, 182, 183] [18, 33, 118, 133, 163, 164] [8, 19, 108, 119, 129, 165] [20, 35, 105, 114, 120, 186] [10, 21, 101, 106, 121, 157] [22, 37, 117, 122, 148, 156] [12, 23, 123, 124, 187, 188] [24, 39, 123, 124, 175, 196] [25, 42, 125, 136, 142, 172] [13, 26, 113, 126, 137, 138] [27, 44, 102, 109, 127, 191] [15, 28, 100, 103, 128, 166] [29, 46, 123, 124, 129, 165] [17, 30, 130, 131, 169, 192] [31, 48, 130, 131, 151, 198] [32, 50, 132, 150, 177, 178] [18, 33, 118, 133, 145, 179] [34, 52, 113, 126, 134, 194] [20, 35, 105, 114, 135, 173] [36, 54, 129, 136, 165, 172] [22, 37, 130, 131, 137, 138] [38, 56, 116, 137, 138, 149] [24, 39, 112, 139, 160, 199] [40, 58, 111, 140, 160, 199] [41, 59, 141, 153, 159, 184] [25, 42, 125, 142, 154, 155] [43, 61, 108, 119, 143, 197] [27, 44, 102, 109, 144, 180] [45, 63, 137, 138, 145, 179] [29, 46, 112, 139, 146, 147] [47, 65, 104, 146, 147, 158] [31, 48, 117, 148, 168, 193] [49, 67, 116, 149, 168, 193] [32, 50, 132, 150, 162, 189] [51, 68, 125, 142, 151, 198] [34, 52, 113, 126, 152, 185] [53, 70, 145, 153, 179, 184] [36, 54, 146, 147, 154, 155] [55, 72, 128, 154, 155, 166] [38, 56, 107, 122, 156, 167] [57, 74, 107, 121, 157, 167] [40, 58, 104, 158, 175, 196] [41, 59, 141, 159, 170, 171] [60, 75, 118, 133, 160, 199] [43, 61, 108, 119, 161, 190] [62, 77, 154, 155, 162, 189] [45, 63, 122, 156, 163, 164] [64, 79, 110, 163, 164, 174] [47, 65, 111, 129, 140, 165] [66, 81, 111, 128, 140, 166] [49, 67, 107, 151, 167, 198] [51, 68, 125, 142, 168, 193] [69, 82, 162, 169, 189, 192] [53, 70, 163, 164, 170, 171] [71, 84, 144, 170, 171, 180] [55, 72, 115, 136, 172, 181] [73, 86, 115, 135, 173, 181] [57, 74, 110, 116, 149, 174] [60, 75, 118, 133, 175, 196] [76, 87, 170, 171, 176, 195] [62, 77, 136, 172, 177, 178] [78, 89, 120, 177, 178, 186] [64, 79, 121, 145, 157, 179] [80, 91, 121, 144, 157, 180] [66, 81, 104, 115, 158, 181] [69, 82, 177, 178, 182, 183] [83, 92, 161, 182, 183, 190] [71, 84, 127, 153, 184, 191] [85, 94, 127, 152, 185, 191] [73, 86, 120, 128, 166, 186] [76, 87, 153, 184, 187, 188] [88, 95, 134, 187, 188, 194] [78, 89, 135, 162, 173, 189] [90, 97, 135, 161, 173, 190] [80, 91, 110, 127, 174, 191] [83, 92, 143, 169, 192, 197] [93, 98, 143, 168, 193, 197] [85, 94, 134, 144, 180, 194] [88, 95, 152, 176, 185, 195] [96, 99, 152, 175, 185, 196] [90, 97, 120, 143, 186, 197] [93, 98, 151, 161, 190, 198] [96, 99, 134, 160, 194, 199]

H_Z (100 checks, sparse supports)

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

Code ID 200-60-2 · download JSON · raw on GitHub