[[160,20,6]] · QEC Challenge

[[160,20,6]] d ≤

160

kd²/n

4.5

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · 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 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[28, 45, 76, 121, 125, 156]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[46, 56, 76, 77, 78, 145]

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 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)×80 (3,2)×160 (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): 80 (2,4): 640 (2,6): 480 (3,2): 160 (3,4): 2640 (3,6): 6960 (3,8): 3600 (3,10): 320

trapping sets H_Z (1,2)×80 (2,2)×80 (3,2)×160 (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): 80 (2,4): 640 (2,6): 480 (3,2): 160 (3,4): 2640 (3,6): 6960 (3,8): 3600 (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

[[160,20,6]] — two-block group-algebra code on SmallGroup(80,25)

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 = 4.50.

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_k20.txt, line 33. This row is SmallGroup(80,25) with nonidentity GAP supports a=[7], b=[5, 8, 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=20), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 6, not an exact certificate. The published distance (d=6) 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(80,25), 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=[7] and b=[5, 8, 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) = 20. 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, 18, 80, 96, 132, 151] [1, 2, 81, 103, 124, 155] [2, 23, 82, 107, 141, 154] [3, 33, 81, 83, 92, 111] [4, 19, 80, 84, 150, 157] [5, 35, 84, 85, 136, 148] [0, 6, 86, 115, 148, 150] [7, 10, 80, 87, 99, 119] [8, 11, 81, 88, 153, 154] [9, 12, 88, 89, 141, 143] [10, 39, 85, 90, 98, 123] [1, 11, 82, 91, 140, 143] [8, 12, 91, 92, 124, 159] [13, 49, 87, 93, 106, 127] [14, 34, 83, 91, 94, 103] [15, 51, 82, 89, 94, 95] [6, 16, 85, 96, 147, 158] [3, 17, 89, 91, 97, 131] [4, 18, 86, 98, 136, 147] [5, 19, 98, 99, 132, 158] [20, 24, 83, 100, 114, 135] [21, 25, 87, 96, 98, 101] [22, 26, 85, 86, 101, 102] [23, 27, 89, 103, 140, 159] [24, 55, 95, 104, 113, 139] [7, 25, 84, 86, 90, 105] [21, 26, 80, 105, 106, 115] [9, 27, 92, 107, 153, 155] [28, 64, 100, 108, 122, 142] [29, 50, 93, 105, 109, 119] [30, 66, 90, 102, 109, 110] [17, 31, 88, 95, 107, 111] [13, 32, 102, 105, 112, 146] [14, 33, 82, 88, 97, 113] [15, 34, 81, 107, 113, 114] [16, 35, 99, 115, 151, 157] [36, 40, 93, 116, 130, 149] [37, 41, 100, 111, 113, 117] [38, 42, 95, 97, 117, 118] [39, 43, 84, 102, 115, 119] [40, 69, 110, 120, 129, 152] [20, 41, 94, 97, 104, 121] [37, 42, 83, 121, 122, 131] [22, 43, 96, 99, 106, 123] [44, 74, 116, 124, 138, 153] [45, 65, 108, 121, 125, 135] [46, 76, 104, 118, 125, 126] [32, 47, 101, 110, 123, 127] [28, 48, 118, 121, 128, 156] [29, 49, 90, 101, 112, 129] [30, 50, 87, 123, 129, 130] [31, 51, 92, 103, 114, 131] [52, 56, 108, 132, 145, 157] [53, 57, 116, 127, 129, 133] [54, 58, 110, 112, 133, 134] [55, 59, 94, 118, 131, 135] [56, 77, 126, 136, 144, 158] [36, 57, 109, 112, 120, 137] [53, 58, 93, 137, 138, 146] [38, 59, 111, 114, 122, 139] [60, 75, 124, 137, 140, 149] [61, 79, 120, 134, 140, 141] [48, 62, 117, 126, 139, 142] [44, 63, 134, 137, 143, 159] [45, 64, 104, 117, 128, 144] [46, 65, 100, 139, 144, 145] [47, 66, 106, 119, 130, 146] [67, 70, 132, 142, 144, 147] [68, 71, 126, 128, 147, 148] [69, 72, 109, 134, 146, 149] [52, 70, 125, 128, 136, 150] [67, 71, 108, 150, 151, 156] [54, 72, 127, 130, 138, 152] [63, 73, 133, 141, 152, 153] [60, 74, 120, 133, 143, 154] [61, 75, 116, 152, 154, 155] [62, 76, 122, 135, 145, 156] [77, 78, 125, 148, 156, 157] [68, 78, 142, 145, 151, 158] [73, 79, 138, 149, 155, 159]

H_Z (80 checks, sparse supports)

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

Code ID 160-20-6 · download JSON · raw on GitHub