[[136,52,2]] · QEC Challenge

[[136,52,2]] d =

136

kd²/n

1.529

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)

[129, 130]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[11, 14]

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 4 · H_Z 4

trapping sets H_X (1,4)×136 (2,0)×102 (3,0)×16 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 136 (2,0): 102 (2,4): 132 (2,6): 1232 (3,0): 16 (3,4): 2680 (3,6): 1472 (3,8): 15408 (3,10): 896

trapping sets H_Z (1,4)×136 (2,0)×102 (3,0)×16 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 136 (2,0): 102 (2,4): 132 (2,6): 1232 (3,0): 16 (3,4): 2680 (3,6): 1472 (3,8): 15408 (3,10): 896

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(68,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order68_k52.txt, line 1. nonidentity supports a=[2, 3, 5], b=[2, 4, 9].

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

[[136,52,2]] — two-block group-algebra code on SmallGroup(68,1)

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 = 1.53.

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_wt8wtL4_order68_k52.txt, line 1. This row is SmallGroup(68,1) with nonidentity GAP supports a=[2, 3, 5], b=[2, 4, 9]. 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=52), 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(68,1), 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, 3, 5], b=[2, 4, 9] 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 = 136 - rank(HX) - rank(HZ) = 52. 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 68 (max weight 8) · Z-checks 68 (max weight 8)

H_X (68 checks, sparse supports)

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

H_Z (68 checks, sparse supports)

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

Code ID 136-52-2 · download JSON · raw on GitHub