[[160,40,4]] · QEC Challenge

[[160,40,4]] d =

160

kd²/n

4.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · 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 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[47, 101, 113, 127]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[56, 60, 69, 145]

certificate exact, d = 4 · scipy/HiGHS cutoff IP

X: no logical < 4 exists; Z: no logical < 4 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 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)×160 (3,2)×720 (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): 160 (2,4): 520 (2,6): 400 (3,2): 720 (3,4): 2400 (3,6): 3680 (3,8): 1760 (3,10): 160

trapping sets H_Z (1,2)×80 (2,2)×160 (3,2)×720 (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): 160 (2,4): 520 (2,6): 400 (3,2): 720 (3,4): 2400 (3,6): 3680 (3,8): 1760 (3,10): 160

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

[[160,40,4]] — two-block group-algebra code on SmallGroup(80,22)

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 = 4.00.

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_wt6wtL2_order80_k40.txt, line 21. This row is SmallGroup(80,22) with nonidentity GAP supports a=[3], b=[3, 8, 22]. 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=40), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 4, not an exact certificate. The published distance (d=4) 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(80,22), 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=[3] and b=[3, 8, 22] 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) = 40. 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, 11, 80, 91, 148, 157] [1, 6, 81, 98, 124, 140] [0, 2, 80, 82, 155, 159] [3, 25, 83, 89, 103, 105] [2, 4, 82, 84, 148, 157] [5, 27, 85, 107, 132, 147] [6, 8, 81, 86, 136, 150] [7, 17, 80, 84, 87, 113] [8, 18, 86, 88, 124, 140] [9, 19, 89, 115, 141, 153] [3, 10, 83, 90, 99, 115] [4, 11, 84, 91, 155, 159] [5, 12, 85, 92, 143, 154] [13, 41, 93, 102, 119, 121] [10, 14, 89, 90, 94, 103] [15, 43, 81, 88, 95, 123] [12, 16, 92, 96, 132, 147] [17, 21, 82, 87, 91, 97] [1, 18, 88, 98, 136, 150] [19, 23, 89, 99, 151, 158] [20, 32, 83, 94, 100, 129] [21, 33, 80, 84, 97, 101] [22, 34, 85, 96, 102, 131] [23, 35, 99, 103, 141, 153] [13, 24, 93, 104, 114, 131] [14, 25, 94, 99, 105, 115] [15, 26, 86, 95, 98, 106] [16, 27, 96, 107, 143, 154] [28, 57, 108, 118, 135, 137] [24, 29, 102, 104, 109, 119] [30, 59, 87, 101, 110, 139] [26, 31, 81, 88, 106, 111] [32, 37, 90, 100, 105, 112] [7, 33, 82, 91, 101, 113] [34, 39, 92, 102, 107, 114] [9, 35, 103, 115, 151, 158] [36, 48, 93, 109, 116, 144] [37, 49, 83, 94, 112, 117] [38, 50, 95, 111, 118, 146] [39, 51, 85, 96, 114, 119] [28, 40, 108, 120, 130, 146] [29, 41, 109, 114, 121, 131] [30, 42, 97, 110, 113, 122] [31, 43, 86, 98, 111, 123] [44, 70, 124, 134, 149, 150] [40, 45, 118, 120, 125, 135] [46, 72, 100, 117, 126, 152] [42, 47, 87, 101, 122, 127] [48, 53, 104, 116, 121, 128] [20, 49, 90, 105, 117, 129] [50, 55, 106, 118, 123, 130] [22, 51, 92, 107, 119, 131] [52, 63, 108, 125, 132, 154] [53, 64, 93, 109, 128, 133] [54, 65, 110, 127, 134, 156] [55, 66, 95, 111, 130, 135] [44, 56, 124, 136, 145, 156] [45, 57, 125, 130, 137, 146] [46, 58, 112, 126, 129, 138] [47, 59, 97, 113, 127, 139] [56, 60, 134, 136, 140, 149] [61, 78, 116, 133, 141, 158] [58, 62, 100, 117, 138, 142] [63, 67, 120, 132, 137, 143] [36, 64, 104, 121, 133, 144] [65, 69, 122, 134, 139, 145] [38, 66, 106, 123, 135, 146] [67, 74, 108, 125, 143, 147] [68, 75, 126, 142, 148, 159] [69, 76, 110, 127, 145, 149] [60, 70, 140, 145, 150, 156] [61, 71, 128, 141, 144, 151] [62, 72, 112, 129, 142, 152] [71, 73, 116, 133, 151, 153] [52, 74, 120, 137, 147, 154] [75, 77, 138, 148, 152, 155] [54, 76, 122, 139, 149, 156] [77, 79, 126, 142, 155, 157] [73, 78, 128, 144, 153, 158] [68, 79, 138, 152, 157, 159]

H_Z (80 checks, sparse supports)

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

Code ID 160-40-4 · download JSON · raw on GitHub