[[160,50,2]] · QEC Challenge

[[160,50,2]] d =

160

kd²/n

1.25

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[122, 130]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[128, 137]

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 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 (2,0)×80 (3,0)×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 (1,4): 80 (2,0): 80 (2,2): 80 (2,4): 480 (2,6): 320 (3,0): 80 (3,2): 280 (3,4): 2360 (3,6): 4000 (3,8): 2240

trapping sets H_Z (1,2)×80 (2,0)×40 (3,2)×640 (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 (2,0): 40 (2,2): 80 (2,4): 720 (3,2): 640 (3,4): 3400 (3,6): 1720 (3,8): 160

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(80,25), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order80_k50.txt, line 1. nonidentity supports a=[2], b=[2, 11, 18].

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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[160,50,2]] — two-block group-algebra code on SmallGroup(80,25)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.25.

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_k50.txt, line 1. This row is SmallGroup(80,25) with nonidentity GAP supports a=[2], b=[2, 11, 18]. 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=50), 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 (d=2) 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(80,25), 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=[2], b=[2, 11, 18] 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) = 50. 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, 1, 80, 81, 136, 154] [0, 1, 80, 81, 143, 150] [2, 6, 82, 98, 124, 132] [3, 7, 82, 83, 87, 98] [4, 8, 84, 103, 155, 158] [5, 9, 85, 89, 151, 159] [2, 6, 86, 91, 124, 132] [3, 7, 83, 86, 87, 91] [4, 8, 88, 96, 151, 159] [5, 9, 85, 89, 155, 158] [10, 17, 80, 81, 90, 113] [11, 18, 86, 91, 147, 153] [12, 19, 92, 115, 141, 148] [13, 20, 90, 93, 100, 113] [14, 21, 94, 99, 107, 119] [15, 22, 92, 95, 102, 115] [16, 23, 88, 96, 143, 150] [10, 17, 80, 81, 97, 105] [11, 18, 82, 98, 140, 157] [12, 19, 99, 107, 141, 148] [13, 20, 93, 97, 100, 105] [14, 21, 92, 101, 111, 115] [15, 22, 95, 99, 102, 107] [16, 23, 84, 103, 136, 154] [24, 32, 83, 87, 104, 129] [25, 33, 88, 96, 97, 105] [26, 34, 85, 89, 106, 131] [27, 35, 99, 107, 140, 157] [28, 36, 104, 108, 116, 129] [29, 37, 109, 114, 123, 135] [30, 38, 106, 110, 118, 131] [31, 39, 86, 91, 101, 111] [24, 32, 83, 87, 112, 121] [25, 33, 84, 90, 103, 113] [26, 34, 85, 89, 114, 123] [27, 35, 92, 115, 147, 153] [28, 36, 108, 112, 116, 121] [29, 37, 106, 117, 127, 131] [30, 38, 110, 114, 118, 123] [31, 39, 82, 94, 98, 119] [40, 48, 93, 100, 120, 144] [41, 49, 101, 111, 112, 121] [42, 50, 95, 102, 122, 146] [43, 51, 84, 103, 114, 123] [44, 52, 120, 124, 132, 144] [45, 53, 125, 130, 139, 149] [46, 54, 122, 126, 134, 146] [47, 55, 97, 105, 117, 127] [40, 48, 93, 100, 128, 137] [41, 49, 94, 104, 119, 129] [42, 50, 95, 102, 130, 139] [43, 51, 88, 96, 106, 131] [44, 52, 124, 128, 132, 137] [45, 53, 122, 133, 142, 146] [46, 54, 126, 130, 134, 139] [47, 55, 90, 109, 113, 135] [56, 63, 108, 116, 136, 154] [57, 64, 117, 127, 128, 137] [58, 65, 110, 118, 138, 156] [59, 66, 94, 119, 130, 139] [60, 67, 140, 145, 152, 157] [61, 68, 138, 141, 148, 156] [62, 69, 112, 121, 133, 142] [56, 63, 108, 116, 143, 150] [57, 64, 109, 120, 135, 144] [58, 65, 110, 118, 145, 152] [59, 66, 101, 111, 122, 146] [60, 67, 138, 147, 153, 156] [61, 68, 141, 145, 148, 152] [62, 69, 104, 125, 129, 149] [70, 74, 133, 142, 143, 150] [71, 75, 126, 134, 151, 159] [72, 76, 109, 135, 145, 152] [73, 77, 128, 137, 147, 153] [70, 74, 125, 136, 149, 154] [71, 75, 126, 134, 155, 158] [72, 76, 117, 127, 138, 156] [73, 77, 120, 140, 144, 157] [78, 79, 125, 149, 155, 158] [78, 79, 133, 142, 151, 159]

H_Z (80 checks, sparse supports)

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

Code ID 160-50-2 · download JSON · raw on GitHub