[[160,6,18]] · QEC Challenge

[[160,6,18]] d ≤

160

kd²/n

12.15

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[5, 26, 30, 35, 46, 55, 61, 71, 86, 88, 99, 115, 119, 125, 126, 133, 143, 148]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[1, 14, 22, 24, 35, 43, 50, 54, 57, 61, 73, 74, 115, 116, 128, 131, 141, 143]

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 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 (2,4): 40 (2,6): 880 (2,10): 1200 (3,2): 80 (3,4): 320 (3,6): 5000 (3,8): 1920 (3,10): 22680 (3,12): 5280 (3,14): 20160 (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 (2,4): 40 (2,6): 880 (2,10): 1200 (3,2): 80 (3,4): 320 (3,6): 5000 (3,8): 1920 (3,10): 22680 (3,12): 5280 (3,14): 20160 (3,16): 1600

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(80,11), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order80_k6.txt, line 71. nonidentity supports a=[13], b=[2, 3, 6, 37, 72].

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

[[160,6,18]] — two-block group-algebra code on SmallGroup(80,11)

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 = 12.15.

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_k6.txt, line 71. This row is SmallGroup(80,11) with nonidentity GAP supports a=[13], b=[2, 3, 6, 37, 72]. 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=6), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 18, 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(80,11), 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, 6, 37, 72] 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) = 6. 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, 72, 80, 87, 91, 102, 120, 127] [1, 34, 80, 81, 98, 108, 128, 135] [2, 47, 80, 82, 94, 97, 114, 139] [3, 78, 81, 83, 104, 106, 116, 141] [4, 59, 82, 84, 89, 100, 105, 142] [5, 11, 80, 85, 88, 106, 136, 148] [6, 9, 81, 82, 86, 101, 120, 146] [7, 48, 83, 87, 95, 112, 114, 148] [8, 19, 84, 86, 88, 94, 113, 149] [9, 50, 81, 89, 93, 114, 127, 143] [10, 61, 83, 85, 86, 90, 128, 151] [11, 62, 84, 91, 99, 108, 112, 152] [0, 12, 82, 85, 92, 98, 109, 155] [13, 71, 88, 90, 92, 93, 101, 153] [14, 24, 83, 94, 100, 120, 122, 135] [2, 15, 81, 84, 92, 95, 121, 157] [16, 26, 85, 96, 122, 133, 149, 150] [17, 21, 87, 89, 90, 97, 106, 155] [18, 22, 88, 91, 98, 105, 116, 156] [19, 23, 86, 89, 99, 104, 117, 139] [20, 33, 85, 93, 97, 99, 100, 157] [21, 63, 84, 87, 101, 128, 130, 141] [22, 35, 83, 88, 99, 102, 129, 142] [23, 65, 89, 103, 111, 130, 153, 154] [24, 73, 93, 95, 98, 104, 113, 158] [3, 25, 90, 94, 96, 105, 112, 146] [4, 26, 86, 91, 95, 106, 124, 159] [5, 27, 92, 96, 107, 125, 144, 156] [10, 28, 87, 93, 105, 107, 108, 149] [29, 40, 94, 109, 119, 136, 138, 157] [12, 30, 95, 107, 110, 135, 137, 147] [31, 42, 96, 111, 117, 134, 138, 158] [32, 36, 92, 100, 102, 104, 112, 159] [33, 37, 91, 97, 101, 103, 113, 151] [34, 38, 90, 98, 102, 114, 132, 152] [35, 39, 99, 103, 115, 123, 133, 158] [36, 49, 80, 100, 113, 115, 116, 153] [37, 74, 101, 117, 125, 142, 143, 145] [38, 51, 102, 115, 118, 126, 141, 144] [39, 76, 96, 103, 119, 140, 145, 159] [13, 40, 97, 104, 108, 110, 120, 156] [14, 41, 105, 109, 111, 121, 131, 159] [15, 42, 106, 110, 122, 140, 146, 154] [16, 43, 107, 111, 123, 129, 141, 145] [25, 44, 108, 121, 123, 124, 134, 148] [45, 56, 103, 109, 125, 147, 150, 152] [27, 46, 110, 119, 123, 126, 132, 151] [47, 58, 101, 104, 111, 118, 127, 152] [48, 52, 82, 112, 116, 118, 128, 158] [49, 53, 113, 117, 119, 129, 137, 152] [50, 54, 114, 118, 130, 138, 147, 151] [51, 55, 107, 115, 119, 131, 148, 150] [52, 64, 116, 127, 129, 131, 132, 140] [53, 79, 109, 117, 126, 133, 154, 156] [54, 66, 110, 118, 125, 131, 134, 155] [18, 55, 85, 112, 119, 124, 135, 156] [28, 56, 120, 124, 126, 136, 145, 155] [29, 57, 115, 121, 125, 127, 137, 154] [30, 58, 122, 126, 131, 138, 143, 153] [31, 59, 83, 113, 123, 127, 130, 139] [41, 60, 118, 124, 133, 137, 139, 140] [61, 70, 89, 91, 125, 132, 141, 158] [43, 62, 90, 103, 116, 126, 139, 142] [63, 67, 128, 132, 134, 139, 143, 150] [64, 68, 121, 129, 133, 135, 138, 144] [65, 69, 122, 130, 134, 137, 145, 157] [1, 66, 87, 92, 131, 135, 136, 146] [67, 75, 111, 124, 132, 144, 146, 147] [32, 68, 94, 98, 110, 133, 148, 159] [6, 69, 95, 97, 109, 134, 146, 149] [44, 70, 130, 136, 140, 142, 144, 150] [45, 71, 80, 99, 137, 141, 143, 151] [46, 72, 93, 115, 128, 138, 142, 152] [57, 73, 82, 102, 117, 140, 151, 153] [74, 77, 123, 136, 143, 147, 149, 154] [7, 75, 81, 105, 122, 144, 148, 155] [8, 76, 100, 106, 121, 145, 149, 156] [17, 77, 86, 96, 108, 147, 155, 157] [60, 78, 84, 114, 129, 150, 153, 158] [20, 79, 88, 107, 120, 154, 157, 159]

H_Z (80 checks, sparse supports)

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

Code ID 160-6-18 · download JSON · raw on GitHub