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[[200,102,2]] d =
n
200
k
102
d
2
kd²/n
2.04
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 8, w_Z = 8 (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)
[137, 151]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[38, 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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×200 (2,0)×200 (3,4)×7100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 200 (2,0): 200 (2,4): 1000 (3,4): 7100 (3,8): 800
trapping sets H_Z (1,4)×200 (2,0)×200 (3,4)×7100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 200 (2,0): 200 (2,4): 1000 (3,4): 7100 (3,8): 800

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(100,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order100_k102.txt, line 1. nonidentity supports a=[2, 3, 6], b=[3, 4, 9].
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 ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[200,102,2]] — two-block group-algebra code on SmallGroup(100,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 2.04.

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_wt8wtL4_order100_k102.txt, line 1. This row is SmallGroup(100,1) with nonidentity GAP supports a=[2, 3, 6], b=[3, 4, 9]. 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=102), 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 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(100,1), 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=[2, 3, 6], b=[3, 4, 9] 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) = 102. 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 8) · Z-checks 100 (max weight 8)
H_X (100 checks, sparse supports)
[0, 1, 2, 5, 100, 102, 155, 168] [0, 1, 2, 5, 101, 105, 164, 175] [0, 1, 2, 5, 100, 102, 155, 168] [3, 6, 8, 13, 100, 102, 103, 108] [4, 7, 9, 14, 104, 109, 172, 182] [0, 1, 2, 5, 101, 105, 164, 175] [3, 6, 8, 13, 101, 105, 106, 113] [4, 7, 9, 14, 107, 114, 179, 187] [3, 6, 8, 13, 100, 102, 103, 108] [4, 7, 9, 14, 104, 109, 172, 182] [10, 15, 18, 25, 103, 108, 110, 118] [11, 16, 19, 26, 104, 109, 111, 119] [12, 17, 20, 27, 112, 120, 185, 192] [3, 6, 8, 13, 101, 105, 106, 113] [4, 7, 9, 14, 107, 114, 179, 187] [10, 15, 18, 25, 106, 113, 115, 125] [11, 16, 19, 26, 107, 114, 116, 126] [12, 17, 20, 27, 117, 127, 190, 195] [10, 15, 18, 25, 103, 108, 110, 118] [11, 16, 19, 26, 104, 109, 111, 119] [12, 17, 20, 27, 112, 120, 185, 192] [21, 28, 32, 41, 110, 118, 121, 132] [22, 29, 33, 42, 111, 119, 122, 133] [23, 30, 34, 43, 112, 120, 123, 134] [24, 31, 35, 44, 124, 135, 194, 198] [10, 15, 18, 25, 106, 113, 115, 125] [11, 16, 19, 26, 107, 114, 116, 126] [12, 17, 20, 27, 117, 127, 190, 195] [21, 28, 32, 41, 115, 125, 128, 141] [22, 29, 33, 42, 116, 126, 129, 142] [23, 30, 34, 43, 117, 127, 130, 143] [24, 31, 35, 44, 131, 144, 197, 199] [21, 28, 32, 41, 110, 118, 121, 132] [22, 29, 33, 42, 111, 119, 122, 133] [23, 30, 34, 43, 112, 120, 123, 134] [24, 31, 35, 44, 124, 135, 194, 198] [36, 45, 50, 59, 121, 132, 136, 150] [37, 46, 51, 60, 122, 133, 137, 151] [38, 47, 52, 61, 123, 134, 138, 152] [39, 48, 53, 62, 124, 135, 139, 153] [40, 49, 54, 63, 136, 140, 150, 154] [21, 28, 32, 41, 115, 125, 128, 141] [22, 29, 33, 42, 116, 126, 129, 142] [23, 30, 34, 43, 117, 127, 130, 143] [24, 31, 35, 44, 131, 144, 197, 199] [36, 45, 50, 59, 128, 141, 145, 159] [37, 46, 51, 60, 129, 142, 146, 160] [38, 47, 52, 61, 130, 143, 147, 161] [39, 48, 53, 62, 131, 144, 148, 162] [40, 49, 54, 63, 145, 149, 159, 163] [36, 45, 50, 59, 121, 132, 136, 150] [37, 46, 51, 60, 122, 133, 137, 151] [38, 47, 52, 61, 123, 134, 138, 152] [39, 48, 53, 62, 124, 135, 139, 153] [40, 49, 54, 63, 136, 140, 150, 154] [55, 64, 68, 75, 137, 151, 155, 168] [56, 65, 69, 76, 138, 152, 156, 169] [57, 66, 70, 77, 139, 153, 157, 170] [58, 67, 71, 78, 140, 154, 158, 171] [36, 45, 50, 59, 128, 141, 145, 159] [37, 46, 51, 60, 129, 142, 146, 160] [38, 47, 52, 61, 130, 143, 147, 161] [39, 48, 53, 62, 131, 144, 148, 162] [40, 49, 54, 63, 145, 149, 159, 163] [55, 64, 68, 75, 146, 160, 164, 175] [56, 65, 69, 76, 147, 161, 165, 176] [57, 66, 70, 77, 148, 162, 166, 177] [58, 67, 71, 78, 149, 163, 167, 178] [55, 64, 68, 75, 137, 151, 155, 168] [56, 65, 69, 76, 138, 152, 156, 169] [57, 66, 70, 77, 139, 153, 157, 170] [58, 67, 71, 78, 140, 154, 158, 171] [72, 79, 82, 87, 156, 169, 172, 182] [73, 80, 83, 88, 157, 170, 173, 183] [74, 81, 84, 89, 158, 171, 174, 184] [55, 64, 68, 75, 146, 160, 164, 175] [56, 65, 69, 76, 147, 161, 165, 176] [57, 66, 70, 77, 148, 162, 166, 177] [58, 67, 71, 78, 149, 163, 167, 178] [72, 79, 82, 87, 165, 176, 179, 187] [73, 80, 83, 88, 166, 177, 180, 188] [74, 81, 84, 89, 167, 178, 181, 189] [72, 79, 82, 87, 156, 169, 172, 182] [73, 80, 83, 88, 157, 170, 173, 183] [74, 81, 84, 89, 158, 171, 174, 184] [85, 90, 92, 95, 173, 183, 185, 192] [86, 91, 93, 96, 174, 184, 186, 193] [72, 79, 82, 87, 165, 176, 179, 187] [73, 80, 83, 88, 166, 177, 180, 188] [74, 81, 84, 89, 167, 178, 181, 189] [85, 90, 92, 95, 180, 188, 190, 195] [86, 91, 93, 96, 181, 189, 191, 196] [85, 90, 92, 95, 173, 183, 185, 192] [86, 91, 93, 96, 174, 184, 186, 193] [94, 97, 98, 99, 186, 193, 194, 198] [85, 90, 92, 95, 180, 188, 190, 195] [86, 91, 93, 96, 181, 189, 191, 196] [94, 97, 98, 99, 191, 196, 197, 199] [94, 97, 98, 99, 186, 193, 194, 198] [94, 97, 98, 99, 191, 196, 197, 199]
H_Z (100 checks, sparse supports)
[0, 2, 3, 8, 100, 101, 102, 105] [1, 5, 6, 13, 100, 101, 102, 105] [0, 2, 3, 8, 100, 101, 102, 105] [3, 8, 10, 18, 103, 106, 108, 113] [4, 9, 11, 19, 104, 107, 109, 114] [1, 5, 6, 13, 100, 101, 102, 105] [6, 13, 15, 25, 103, 106, 108, 113] [7, 14, 16, 26, 104, 107, 109, 114] [3, 8, 10, 18, 103, 106, 108, 113] [4, 9, 11, 19, 104, 107, 109, 114] [10, 18, 21, 32, 110, 115, 118, 125] [11, 19, 22, 33, 111, 116, 119, 126] [12, 20, 23, 34, 112, 117, 120, 127] [6, 13, 15, 25, 103, 106, 108, 113] [7, 14, 16, 26, 104, 107, 109, 114] [15, 25, 28, 41, 110, 115, 118, 125] [16, 26, 29, 42, 111, 116, 119, 126] [17, 27, 30, 43, 112, 117, 120, 127] [10, 18, 21, 32, 110, 115, 118, 125] [11, 19, 22, 33, 111, 116, 119, 126] [12, 20, 23, 34, 112, 117, 120, 127] [21, 32, 36, 50, 121, 128, 132, 141] [22, 33, 37, 51, 122, 129, 133, 142] [23, 34, 38, 52, 123, 130, 134, 143] [24, 35, 39, 53, 124, 131, 135, 144] [15, 25, 28, 41, 110, 115, 118, 125] [16, 26, 29, 42, 111, 116, 119, 126] [17, 27, 30, 43, 112, 117, 120, 127] [28, 41, 45, 59, 121, 128, 132, 141] [29, 42, 46, 60, 122, 129, 133, 142] [30, 43, 47, 61, 123, 130, 134, 143] [31, 44, 48, 62, 124, 131, 135, 144] [21, 32, 36, 50, 121, 128, 132, 141] [22, 33, 37, 51, 122, 129, 133, 142] [23, 34, 38, 52, 123, 130, 134, 143] [24, 35, 39, 53, 124, 131, 135, 144] [36, 40, 50, 54, 136, 145, 150, 159] [37, 51, 55, 68, 137, 146, 151, 160] [38, 52, 56, 69, 138, 147, 152, 161] [39, 53, 57, 70, 139, 148, 153, 162] [40, 54, 58, 71, 140, 149, 154, 163] [28, 41, 45, 59, 121, 128, 132, 141] [29, 42, 46, 60, 122, 129, 133, 142] [30, 43, 47, 61, 123, 130, 134, 143] [31, 44, 48, 62, 124, 131, 135, 144] [45, 49, 59, 63, 136, 145, 150, 159] [46, 60, 64, 75, 137, 146, 151, 160] [47, 61, 65, 76, 138, 147, 152, 161] [48, 62, 66, 77, 139, 148, 153, 162] [49, 63, 67, 78, 140, 149, 154, 163] [36, 40, 50, 54, 136, 145, 150, 159] [37, 51, 55, 68, 137, 146, 151, 160] [38, 52, 56, 69, 138, 147, 152, 161] [39, 53, 57, 70, 139, 148, 153, 162] [40, 54, 58, 71, 140, 149, 154, 163] [0, 2, 55, 68, 155, 164, 168, 175] [56, 69, 72, 82, 156, 165, 169, 176] [57, 70, 73, 83, 157, 166, 170, 177] [58, 71, 74, 84, 158, 167, 171, 178] [45, 49, 59, 63, 136, 145, 150, 159] [46, 60, 64, 75, 137, 146, 151, 160] [47, 61, 65, 76, 138, 147, 152, 161] [48, 62, 66, 77, 139, 148, 153, 162] [49, 63, 67, 78, 140, 149, 154, 163] [1, 5, 64, 75, 155, 164, 168, 175] [65, 76, 79, 87, 156, 165, 169, 176] [66, 77, 80, 88, 157, 166, 170, 177] [67, 78, 81, 89, 158, 167, 171, 178] [0, 2, 55, 68, 155, 164, 168, 175] [56, 69, 72, 82, 156, 165, 169, 176] [57, 70, 73, 83, 157, 166, 170, 177] [58, 71, 74, 84, 158, 167, 171, 178] [4, 9, 72, 82, 172, 179, 182, 187] [73, 83, 85, 92, 173, 180, 183, 188] [74, 84, 86, 93, 174, 181, 184, 189] [1, 5, 64, 75, 155, 164, 168, 175] [65, 76, 79, 87, 156, 165, 169, 176] [66, 77, 80, 88, 157, 166, 170, 177] [67, 78, 81, 89, 158, 167, 171, 178] [7, 14, 79, 87, 172, 179, 182, 187] [80, 88, 90, 95, 173, 180, 183, 188] [81, 89, 91, 96, 174, 181, 184, 189] [4, 9, 72, 82, 172, 179, 182, 187] [73, 83, 85, 92, 173, 180, 183, 188] [74, 84, 86, 93, 174, 181, 184, 189] [12, 20, 85, 92, 185, 190, 192, 195] [86, 93, 94, 98, 186, 191, 193, 196] [7, 14, 79, 87, 172, 179, 182, 187] [80, 88, 90, 95, 173, 180, 183, 188] [81, 89, 91, 96, 174, 181, 184, 189] [17, 27, 90, 95, 185, 190, 192, 195] [91, 96, 97, 99, 186, 191, 193, 196] [12, 20, 85, 92, 185, 190, 192, 195] [86, 93, 94, 98, 186, 191, 193, 196] [24, 35, 94, 98, 194, 197, 198, 199] [17, 27, 90, 95, 185, 190, 192, 195] [91, 96, 97, 99, 186, 191, 193, 196] [31, 44, 97, 99, 194, 197, 198, 199] [24, 35, 94, 98, 194, 197, 198, 199] [31, 44, 97, 99, 194, 197, 198, 199]
Code ID 200-102-2 · download JSON · raw on GitHub