[[128,64,2]] · QEC Challenge

[[128,64,2]] d =

128

kd²/n

2.0

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)

[11, 31]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[9, 29]

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 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)

trapping sets H_X (1,2)×64 (2,0)×64 (3,2)×960 (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 (2,0): 64 (2,4): 384 (2,8): 384 (3,2): 960 (3,6): 3904 (3,10): 3072 (3,14): 256

trapping sets H_Z (1,2)×64 (2,0)×64 (3,2)×960 (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 (2,0): 64 (2,4): 384 (2,8): 384 (3,2): 960 (3,6): 3904 (3,10): 3072 (3,14): 256

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(64,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order64_k64.txt, line 1. nonidentity supports a=[6], b=[2, 3, 6, 11, 15].

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

[[128,64,2]] — two-block group-algebra code on SmallGroup(64,3)

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 = 2.00.

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_k64.txt, line 1. This row is SmallGroup(64,3) with nonidentity GAP supports a=[6], b=[2, 3, 6, 11, 15]. 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=64), 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(64,3), 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=[6], b=[2, 3, 6, 11, 15] 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) = 64. 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, 5, 64, 69, 72, 91, 100, 119] [1, 10, 64, 65, 69, 74, 110, 124] [2, 14, 64, 66, 69, 78, 86, 107] [3, 17, 65, 67, 74, 81, 117, 126] [4, 19, 66, 68, 78, 83, 90, 112] [0, 5, 64, 69, 72, 91, 100, 119] [6, 21, 70, 77, 85, 92, 99, 114] [7, 24, 65, 66, 71, 74, 78, 88] [8, 27, 67, 72, 81, 91, 122, 127] [9, 29, 68, 71, 73, 83, 88, 93] [1, 10, 64, 65, 69, 74, 110, 124] [11, 31, 70, 75, 85, 87, 95, 109] [12, 33, 67, 71, 76, 81, 88, 97] [13, 35, 68, 77, 83, 99, 106, 121] [2, 14, 64, 66, 69, 78, 86, 107] [15, 37, 70, 79, 85, 101, 108, 123] [16, 38, 73, 76, 80, 93, 97, 102] [3, 17, 65, 67, 74, 81, 117, 126] [18, 40, 75, 82, 95, 96, 104, 116] [4, 19, 66, 68, 78, 83, 90, 112] [20, 41, 79, 84, 101, 105, 113, 125] [6, 21, 70, 77, 85, 92, 99, 114] [22, 43, 72, 76, 86, 91, 97, 107] [23, 45, 73, 77, 87, 93, 99, 109] [7, 24, 65, 66, 71, 74, 78, 88] [25, 47, 75, 79, 89, 95, 101, 111] [26, 48, 80, 86, 90, 102, 107, 112] [8, 27, 67, 72, 81, 91, 122, 127] [28, 50, 82, 92, 104, 106, 114, 121] [9, 29, 68, 71, 73, 83, 88, 93] [30, 51, 84, 89, 94, 105, 111, 115] [11, 31, 70, 75, 85, 87, 95, 109] [32, 52, 80, 87, 96, 102, 109, 116] [12, 33, 67, 71, 76, 81, 88, 97] [34, 54, 82, 89, 98, 104, 111, 118] [13, 35, 68, 77, 83, 99, 106, 121] [36, 55, 84, 100, 105, 119, 122, 127] [15, 37, 70, 79, 85, 101, 108, 123] [16, 38, 73, 76, 80, 93, 97, 102] [39, 56, 94, 98, 103, 115, 118, 120] [18, 40, 75, 82, 95, 96, 104, 116] [20, 41, 79, 84, 101, 105, 113, 125] [42, 57, 90, 96, 106, 112, 116, 121] [22, 43, 72, 76, 86, 91, 97, 107] [44, 59, 92, 98, 108, 114, 118, 123] [23, 45, 73, 77, 87, 93, 99, 109] [46, 60, 94, 100, 110, 115, 119, 124] [25, 47, 75, 79, 89, 95, 101, 111] [26, 48, 80, 86, 90, 102, 107, 112] [49, 61, 103, 108, 113, 120, 123, 125] [28, 50, 82, 92, 104, 106, 114, 121] [30, 51, 84, 89, 94, 105, 111, 115] [32, 52, 80, 87, 96, 102, 109, 116] [53, 62, 103, 110, 117, 120, 124, 126] [34, 54, 82, 89, 98, 104, 111, 118] [36, 55, 84, 100, 105, 119, 122, 127] [39, 56, 94, 98, 103, 115, 118, 120] [42, 57, 90, 96, 106, 112, 116, 121] [58, 63, 113, 117, 122, 125, 126, 127] [44, 59, 92, 98, 108, 114, 118, 123] [46, 60, 94, 100, 110, 115, 119, 124] [49, 61, 103, 108, 113, 120, 123, 125] [53, 62, 103, 110, 117, 120, 124, 126] [58, 63, 113, 117, 122, 125, 126, 127]

H_Z (64 checks, sparse supports)

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

Code ID 128-64-2 · download JSON · raw on GitHub