[[160,80,2]] · QEC Challenge

[[160,80,2]] d =

160

kd²/n

2.0

X/Z

Share this result

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)

[5, 15]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[45, 60]

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 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)

trapping sets H_X (1,2)×80 (2,0)×80 (3,2)×1520 (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 (2,0): 80 (2,4): 560 (2,8): 320 (3,2): 1520 (3,6): 4160 (3,10): 2560

trapping sets H_Z (1,2)×80 (2,0)×80 (3,2)×1520 (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 (2,0): 80 (2,4): 560 (2,8): 320 (3,2): 1520 (3,6): 4160 (3,10): 2560

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(80,4), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order80_k80.txt, line 1. nonidentity supports a=[5], b=[2, 5, 9, 13, 27].

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

[[160,80,2]] — two-block group-algebra code on SmallGroup(80,4)

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 = 2.00.

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_k80.txt, line 1. This row is SmallGroup(80,4) with nonidentity GAP supports a=[5], b=[2, 5, 9, 13, 27]. 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=80), 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(80,4), 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=[5], b=[2, 5, 9, 13, 27] 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) = 80. 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, 4, 80, 81, 84, 88, 151, 158] [1, 8, 80, 81, 84, 88, 155, 159] [2, 11, 82, 86, 91, 98, 127, 142] [3, 13, 83, 87, 93, 100, 139, 152] [0, 4, 80, 81, 84, 88, 151, 158] [5, 15, 85, 90, 95, 104, 135, 149] [6, 18, 82, 86, 91, 98, 135, 149] [7, 20, 83, 87, 93, 100, 146, 156] [1, 8, 80, 81, 84, 88, 155, 159] [9, 22, 89, 97, 102, 112, 127, 142] [10, 24, 90, 97, 104, 112, 141, 153] [2, 11, 82, 86, 91, 98, 127, 142] [12, 26, 80, 84, 92, 106, 146, 156] [3, 13, 83, 87, 93, 100, 139, 152] [14, 28, 82, 91, 94, 108, 148, 157] [5, 15, 85, 90, 95, 104, 135, 149] [16, 30, 96, 105, 110, 119, 120, 134] [17, 32, 90, 97, 104, 112, 148, 157] [6, 18, 82, 86, 91, 98, 135, 149] [19, 34, 81, 88, 99, 114, 139, 152] [7, 20, 83, 87, 93, 100, 146, 156] [21, 36, 86, 98, 101, 116, 141, 153] [9, 22, 89, 97, 102, 112, 127, 142] [23, 38, 103, 111, 113, 118, 126, 128] [10, 24, 90, 97, 104, 112, 141, 153] [25, 40, 83, 93, 105, 120, 155, 159] [12, 26, 80, 84, 92, 106, 146, 156] [27, 42, 85, 95, 107, 122, 131, 145] [14, 28, 82, 91, 94, 108, 148, 157] [29, 44, 92, 106, 109, 124, 133, 147] [16, 30, 96, 105, 110, 119, 120, 134] [31, 46, 103, 111, 118, 121, 126, 136] [17, 32, 90, 97, 104, 112, 148, 157] [33, 48, 87, 100, 113, 128, 151, 158] [19, 34, 81, 88, 99, 114, 139, 152] [35, 50, 89, 102, 115, 123, 130, 138] [21, 36, 86, 98, 101, 116, 141, 153] [37, 52, 99, 114, 117, 125, 132, 140] [23, 38, 103, 111, 113, 118, 126, 128] [39, 54, 96, 110, 119, 129, 134, 143] [25, 40, 83, 93, 105, 120, 155, 159] [41, 56, 94, 108, 121, 136, 144, 154] [27, 42, 85, 95, 107, 122, 131, 145] [43, 58, 96, 110, 115, 123, 130, 138] [29, 44, 92, 106, 109, 124, 133, 147] [45, 60, 107, 117, 122, 125, 132, 140] [31, 46, 103, 111, 118, 121, 126, 136] [47, 62, 89, 102, 127, 137, 142, 150] [33, 48, 87, 100, 113, 128, 151, 158] [49, 63, 101, 116, 129, 137, 143, 150] [35, 50, 89, 102, 115, 123, 130, 138] [51, 65, 103, 107, 118, 122, 131, 145] [37, 52, 99, 114, 117, 125, 132, 140] [53, 67, 109, 115, 124, 130, 133, 147] [39, 54, 96, 110, 119, 129, 134, 143] [55, 69, 85, 95, 135, 144, 149, 154] [41, 56, 94, 108, 121, 136, 144, 154] [57, 70, 109, 124, 129, 137, 143, 150] [43, 58, 96, 110, 115, 123, 130, 138] [59, 72, 99, 111, 114, 126, 139, 152] [45, 60, 107, 117, 122, 125, 132, 140] [61, 73, 101, 116, 123, 138, 141, 153] [47, 62, 89, 102, 127, 137, 142, 150] [49, 63, 101, 116, 129, 137, 143, 150] [64, 74, 117, 121, 132, 136, 144, 154] [51, 65, 103, 107, 118, 122, 131, 145] [66, 76, 92, 106, 119, 134, 146, 156] [53, 67, 109, 115, 124, 130, 133, 147] [68, 77, 94, 108, 131, 145, 148, 157] [55, 69, 85, 95, 135, 144, 149, 154] [57, 70, 109, 124, 129, 137, 143, 150] [71, 78, 113, 125, 128, 140, 151, 158] [59, 72, 99, 111, 114, 126, 139, 152] [61, 73, 101, 116, 123, 138, 141, 153] [64, 74, 117, 121, 132, 136, 144, 154] [75, 79, 105, 120, 133, 147, 155, 159] [66, 76, 92, 106, 119, 134, 146, 156] [68, 77, 94, 108, 131, 145, 148, 157] [71, 78, 113, 125, 128, 140, 151, 158] [75, 79, 105, 120, 133, 147, 155, 159]

H_Z (80 checks, sparse supports)

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

Code ID 160-80-2 · download JSON · raw on GitHub