[[84,24,2]] · QEC Challenge

[[84,24,2]] d =

kd²/n

1.143

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 7, w_Z = 7 (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)

[30, 33]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[68, 71]

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 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×42 (2,0)×21 (3,0)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 42 (1,4): 42 (2,0): 21 (2,4): 210 (2,5): 504 (3,0): 56 (3,3): 420 (3,4): 84 (3,5): 1008 (3,6): 4284 (3,7): 2562 (3,8): 504

trapping sets H_Z (1,3)×42 (2,0)×21 (3,0)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 42 (1,4): 42 (2,0): 21 (2,4): 210 (2,5): 504 (3,0): 56 (3,3): 420 (3,4): 84 (3,5): 1008 (3,6): 4284 (3,7): 2562 (3,8): 504

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(42,6), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order42_k24.txt, line 1. nonidentity supports a=[4, 15], b=[2, 3, 5].

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

[[84,24,2]] — two-block group-algebra code on SmallGroup(42,6)

Direction & hypothesis

Target: unrestricted weight-8 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 = 1.14.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order42_k24.txt, line 1. This row is SmallGroup(42,6) with nonidentity GAP supports a=[4, 15], b=[2, 3, 5]. 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=24), 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(42,6), 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=[4, 15], b=[2, 3, 5] 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 = 84 - rank(HX) - rank(HZ) = 24. 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 42 (max weight 7) · Z-checks 42 (max weight 7)

H_X (42 checks, sparse supports)

[0, 20, 32, 42, 43, 48, 51] [1, 23, 35, 42, 43, 48, 51] [2, 25, 37, 42, 43, 44, 46] [0, 3, 26, 45, 47, 54, 57] [4, 28, 39, 42, 43, 44, 46] [1, 5, 29, 45, 47, 54, 57] [6, 30, 40, 44, 46, 48, 51] [2, 7, 31, 45, 47, 49, 52] [3, 8, 32, 50, 53, 60, 63] [9, 33, 41, 44, 46, 48, 51] [4, 10, 34, 45, 47, 49, 52] [5, 11, 35, 50, 53, 60, 63] [6, 12, 36, 49, 52, 54, 57] [7, 13, 37, 50, 53, 55, 58] [0, 8, 14, 56, 59, 66, 69] [9, 15, 38, 49, 52, 54, 57] [10, 16, 39, 50, 53, 55, 58] [1, 11, 17, 56, 59, 66, 69] [12, 18, 40, 55, 58, 60, 63] [2, 13, 19, 56, 59, 61, 64] [3, 14, 20, 62, 65, 72, 75] [15, 21, 41, 55, 58, 60, 63] [4, 16, 22, 56, 59, 61, 64] [5, 17, 23, 62, 65, 72, 75] [6, 18, 24, 61, 64, 66, 69] [7, 19, 25, 62, 65, 67, 70] [8, 20, 26, 68, 71, 78, 80] [9, 21, 27, 61, 64, 66, 69] [10, 22, 28, 62, 65, 67, 70] [11, 23, 29, 68, 71, 78, 80] [12, 24, 30, 67, 70, 72, 75] [13, 25, 31, 68, 71, 73, 76] [14, 26, 32, 74, 77, 82, 83] [15, 27, 33, 67, 70, 72, 75] [16, 28, 34, 68, 71, 73, 76] [17, 29, 35, 74, 77, 82, 83] [18, 30, 36, 73, 76, 78, 80] [19, 31, 37, 74, 77, 79, 81] [21, 33, 38, 73, 76, 78, 80] [22, 34, 39, 74, 77, 79, 81] [24, 36, 40, 79, 81, 82, 83] [27, 38, 41, 79, 81, 82, 83]

H_Z (42 checks, sparse supports)

[0, 1, 2, 4, 42, 45, 56] [0, 1, 2, 4, 43, 47, 59] [2, 4, 6, 9, 44, 49, 61] [3, 5, 7, 10, 45, 50, 62] [2, 4, 6, 9, 46, 52, 64] [3, 5, 7, 10, 47, 53, 65] [0, 1, 6, 9, 48, 54, 66] [7, 10, 12, 15, 49, 55, 67] [8, 11, 13, 16, 50, 56, 68] [0, 1, 6, 9, 51, 57, 69] [7, 10, 12, 15, 52, 58, 70] [8, 11, 13, 16, 53, 59, 71] [3, 5, 12, 15, 54, 60, 72] [13, 16, 18, 21, 55, 61, 73] [14, 17, 19, 22, 56, 62, 74] [3, 5, 12, 15, 57, 63, 75] [13, 16, 18, 21, 58, 64, 76] [14, 17, 19, 22, 59, 65, 77] [8, 11, 18, 21, 60, 66, 78] [19, 22, 24, 27, 61, 67, 79] [20, 23, 25, 28, 42, 62, 68] [8, 11, 18, 21, 63, 69, 80] [19, 22, 24, 27, 64, 70, 81] [20, 23, 25, 28, 43, 65, 71] [14, 17, 24, 27, 66, 72, 82] [25, 28, 30, 33, 44, 67, 73] [26, 29, 31, 34, 45, 68, 74] [14, 17, 24, 27, 69, 75, 83] [25, 28, 30, 33, 46, 70, 76] [26, 29, 31, 34, 47, 71, 77] [20, 23, 30, 33, 48, 72, 78] [31, 34, 36, 38, 49, 73, 79] [32, 35, 37, 39, 42, 50, 74] [20, 23, 30, 33, 51, 75, 80] [31, 34, 36, 38, 52, 76, 81] [32, 35, 37, 39, 43, 53, 77] [26, 29, 36, 38, 54, 78, 82] [37, 39, 40, 41, 44, 55, 79] [26, 29, 36, 38, 57, 80, 83] [37, 39, 40, 41, 46, 58, 81] [32, 35, 40, 41, 48, 60, 82] [32, 35, 40, 41, 51, 63, 83]

Code ID 84-24-2 · download JSON · raw on GitHub