[[128,6,16]] · QEC Challenge

[[128,6,16]] d ≤

128

kd²/n

12.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (claimed upper_bound)

witness operator (support, 16 qubits)

[9, 21, 24, 35, 36, 47, 59, 62, 64, 74, 89, 93, 94, 95, 125, 126]

d_Z 16 · witness weight 16 (claimed upper_bound)

witness operator (support, 16 qubits)

[14, 15, 18, 30, 31, 39, 44, 45, 49, 52, 55, 58, 89, 101, 104, 122]

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 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)×64 (2,2)×64 (3,2)×64 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,2): 64 (1,6): 64 (2,2): 64 (2,6): 768 (2,8): 128 (2,10): 704 (3,2): 64 (3,4): 64 (3,6): 4416 (3,8): 2816 (3,10): 16384 (3,12): 6080 (3,14): 8704 (3,16): 512

trapping sets H_Z (1,2)×64 (2,2)×64 (3,2)×64 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,2): 64 (1,6): 64 (2,2): 64 (2,6): 768 (2,8): 128 (2,10): 704 (3,2): 64 (3,4): 64 (3,6): 4416 (3,8): 2816 (3,10): 16384 (3,12): 6080 (3,14): 8704 (3,16): 512

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(64,27), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order64_k6.txt, line 25. nonidentity supports a=[2], b=[3, 4, 10, 22, 36].

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

[[128,6,16]] — two-block group-algebra code on SmallGroup(64,27)

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 = 12.00.

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_order64_k6.txt, line 25. This row is SmallGroup(64,27) with nonidentity GAP supports a=[2], b=[3, 4, 10, 22, 36]. 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=6), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 16, 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(64,27), 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=[3, 4, 10, 22, 36] 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 = 128 - rank(HX) - rank(HZ) = 6. 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 64 (max weight 8) · Z-checks 64 (max weight 8)

H_X (64 checks, sparse supports)

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

H_Z (64 checks, sparse supports)

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

Code ID 128-6-16 · download JSON · raw on GitHub