[[160,4,19]] · QEC Challenge

[[160,4,19]] d ≤

160

kd²/n

9.025

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · 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 19 · witness weight 19 (claimed upper_bound)

witness operator (support, 19 qubits)

[1, 46, 77, 80, 81, 87, 91, 94, 99, 100, 113, 132, 134, 142, 146, 148, 149, 154, 156]

d_Z 19 · witness weight 19 (claimed upper_bound)

witness operator (support, 19 qubits)

[0, 5, 22, 29, 32, 39, 49, 50, 51, 54, 55, 63, 64, 66, 71, 77, 85, 130, 154]

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 8 · H_Z 8

qubit degrees H_X 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)

trapping sets H_X (1,2)×80 (2,2)×80 (3,2)×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): 80 (1,6): 80 (2,2): 80 (2,6): 960 (2,10): 1200 (3,2): 80 (3,6): 5760 (3,8): 1440 (3,10): 24000 (3,12): 5840 (3,14): 19680 (3,16): 1600

trapping sets H_Z (1,2)×80 (2,2)×80 (3,2)×80 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 80 (1,6): 80 (2,2): 80 (2,6): 960 (2,10): 1200 (3,2): 80 (3,6): 5760 (3,8): 1440 (3,10): 24000 (3,12): 5840 (3,14): 19680 (3,16): 1600

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(80,8), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order80_k4.txt, line 75. nonidentity supports a=[13], b=[2, 3, 15, 38, 73].

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

[[160,4,19]] — two-block group-algebra code on SmallGroup(80,8)

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 = 9.03.

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_wt8wtL2_order80_k4.txt, line 75. This row is SmallGroup(80,8) with nonidentity GAP supports a=[13], b=[2, 3, 15, 38, 73]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 19, 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(80,8), 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=[13], b=[2, 3, 15, 38, 73] 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 = 160 - rank(HX) - rank(HZ) = 4. 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 80 (max weight 8) · Z-checks 80 (max weight 8)

H_X (80 checks, sparse supports)

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

H_Z (80 checks, sparse supports)

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

Code ID 160-4-19 · download JSON · raw on GitHub