[[164,2,20]] · QEC Challenge

[[164,2,20]] d ≤

164

kd²/n

4.878

X/Z

Share this result

Distance

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

witness operator (support, 20 qubits)

[9, 14, 17, 22, 23, 30, 32, 57, 62, 64, 83, 87, 98, 102, 103, 137, 143, 145, 154, 163]

d_Z 20 · witness weight 20 (claimed upper_bound)

witness operator (support, 20 qubits)

[10, 13, 14, 17, 19, 20, 23, 25, 46, 68, 97, 102, 108, 112, 118, 122, 130, 134, 141, 146]

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 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)×164 (2,4)×205 (3,4)×328 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 164 (2,4): 205 (2,6): 1886 (3,4): 328 (3,6): 5330 (3,8): 31488 (3,10): 2624

trapping sets H_Z (1,4)×164 (2,4)×205 (3,4)×328 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 164 (2,4): 205 (2,6): 1886 (3,4): 328 (3,6): 5330 (3,8): 31488 (3,10): 2624

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(82,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order82_k2.txt, line 15. nonidentity supports a=[2, 3, 5], b=[11, 33, 45].

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

[[164,2,20]] — two-block group-algebra code on SmallGroup(82,1)

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 = 4.88.

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_order82_k2.txt, line 15. This row is SmallGroup(82,1) with nonidentity GAP supports a=[2, 3, 5], b=[11, 33, 45]. 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 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(82,1), 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, 5], b=[11, 33, 45] 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 = 164 - 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 82 (max weight 8) · Z-checks 82 (max weight 8)

H_X (82 checks, sparse supports)

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

H_Z (82 checks, sparse supports)

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

Code ID 164-2-20 · download JSON · raw on GitHub