[[160,2,17]] · QEC Challenge

[[160,2,17]] d ≤

160

kd²/n

3.612

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · 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 17 · witness weight 17 (claimed upper_bound)

witness operator (support, 17 qubits)

[24, 27, 50, 70, 80, 85, 90, 94, 95, 96, 107, 121, 122, 126, 140, 154, 158]

d_Z 17 · witness weight 17 (claimed upper_bound)

witness operator (support, 17 qubits)

[2, 3, 5, 24, 25, 27, 30, 33, 44, 47, 58, 59, 70, 73, 75, 94, 134]

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 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)×80 (2,2)×120 (3,2)×240 (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 (1,4): 80 (2,2): 120 (2,4): 560 (2,6): 480 (3,2): 240 (3,4): 2520 (3,6): 6200 (3,8): 4560 (3,10): 320

trapping sets H_Z (1,2)×80 (2,2)×120 (3,2)×240 (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 (1,4): 80 (2,2): 120 (2,4): 560 (2,6): 480 (3,2): 240 (3,4): 2520 (3,6): 6200 (3,8): 4560 (3,10): 320

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-18

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

[[160,2,17]] — two-block group-algebra code on SmallGroup(80,1)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 3.61.

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_order80_k2.txt, line 5. This row is SmallGroup(80,1) with nonidentity GAP supports a=[13], b=[7, 17, 71]. 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 ≤ 17, not an exact certificate. The published distance (d=17) 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

Union Alpha (maker currently anonymous), 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(80,1), 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=[13] and b=[7, 17, 71] 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) = 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 80 (max weight 6) · Z-checks 80 (max weight 6)

H_X (80 checks, sparse supports)

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

H_Z (80 checks, sparse supports)

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

Code ID 160-2-17 · download JSON · raw on GitHub