[[96,20,6]] · QEC Challenge

[[96,20,6]] d =

kd²/n

7.5

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · 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 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[51, 52, 61, 64, 76, 78]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[69, 73, 74, 81, 85, 92]

certificate exact, d = 6 · scipy/HiGHS MILP

X: no logical < 6 exists; Z: no logical < 6 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 6 · H_Z 6 (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)×48 (2,4)×144 (3,3)×48 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 48 (1,4): 48 (2,4): 144 (2,5): 576 (2,6): 288 (3,3): 48 (3,5): 432 (3,6): 5472 (3,7): 7536 (3,8): 1440 (3,9): 864 (3,10): 192

trapping sets H_Z (1,3)×48 (2,4)×144 (3,3)×48 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 48 (1,4): 48 (2,4): 144 (2,5): 576 (2,6): 288 (3,3): 48 (3,5): 432 (3,6): 5472 (3,7): 7536 (3,8): 1440 (3,9): 864 (3,10): 192

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-18

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

[[96,20,6]] — two-block group-algebra code on SmallGroup(48,20)

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 = 20·36/96 = 7.50.

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_order48_k20.txt, line 2. This row is SmallGroup(48,20) with nonidentity GAP supports a=[15,30], b=[2,3,35]. 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=20), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 6, not an exact certificate. The published distance (d=6) 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

Union Alpha (maker currently anonymous), 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(48,20), 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=[15,30] and b=[2,3,35] 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 = 96 - rank(HX) - rank(HZ) = 20. 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 48 (max weight 7) · Z-checks 48 (max weight 7)

H_X (48 checks, sparse supports)

[0, 15, 28, 48, 54, 56, 60] [1, 22, 35, 48, 49, 59, 67] [2, 26, 38, 48, 50, 57, 66] [3, 4, 29, 51, 65, 69, 74] [4, 13, 30, 49, 52, 66, 75] [3, 5, 41, 50, 53, 67, 71] [6, 33, 42, 49, 50, 54, 64] [7, 8, 36, 51, 55, 73, 81] [8, 20, 37, 50, 52, 56, 82] [7, 9, 45, 53, 54, 57, 75] [10, 11, 39, 51, 58, 70, 80] [11, 24, 40, 52, 54, 59, 71] [10, 12, 46, 49, 53, 60, 82] [5, 13, 14, 61, 79, 83, 87] [0, 14, 41, 55, 62, 80, 88] [13, 15, 16, 58, 63, 81, 85] [14, 16, 29, 57, 59, 64, 82] [17, 18, 43, 55, 58, 65, 78] [18, 31, 44, 53, 56, 59, 66] [17, 19, 47, 52, 57, 60, 67] [9, 20, 21, 61, 68, 86, 91] [1, 21, 45, 58, 62, 69, 92] [20, 22, 23, 63, 65, 70, 88] [21, 23, 36, 60, 64, 66, 71] [12, 24, 25, 61, 72, 84, 90] [2, 25, 46, 62, 65, 73, 85] [24, 26, 27, 55, 63, 74, 92] [25, 27, 39, 56, 64, 67, 75] [3, 16, 28, 68, 76, 90, 94] [0, 29, 30, 72, 77, 91, 93] [5, 28, 30, 70, 73, 78, 92] [19, 31, 32, 68, 72, 79, 89] [6, 32, 47, 63, 69, 73, 80] [31, 33, 34, 62, 70, 74, 81] [32, 34, 43, 48, 71, 75, 82] [7, 23, 35, 72, 76, 83, 95] [1, 36, 37, 77, 79, 84, 94] [9, 35, 37, 74, 78, 80, 85] [10, 27, 38, 76, 79, 86, 93] [2, 39, 40, 68, 77, 87, 95] [12, 38, 40, 69, 78, 81, 88] [4, 15, 41, 84, 86, 89, 95] [17, 34, 42, 77, 83, 86, 90] [6, 43, 44, 76, 84, 87, 91] [19, 42, 44, 51, 85, 88, 92] [8, 22, 45, 87, 89, 90, 93] [11, 26, 46, 83, 89, 91, 94] [18, 33, 47, 61, 93, 94, 95]

H_Z (48 checks, sparse supports)

[0, 1, 2, 34, 48, 62, 77] [1, 4, 6, 12, 49, 69, 84] [2, 5, 6, 8, 50, 73, 87] [3, 7, 10, 44, 51, 53, 76] [4, 8, 11, 19, 51, 52, 89] [5, 9, 12, 18, 53, 61, 78] [0, 6, 9, 11, 54, 80, 91] [7, 14, 17, 26, 55, 57, 83] [0, 8, 18, 27, 55, 56, 93] [2, 9, 16, 19, 57, 68, 85] [10, 15, 17, 21, 58, 60, 86] [1, 11, 16, 18, 58, 59, 94] [0, 12, 19, 23, 60, 72, 88] [13, 20, 24, 47, 52, 61, 63] [14, 21, 25, 33, 61, 62, 64] [15, 22, 26, 32, 48, 63, 89] [6, 16, 23, 27, 63, 64, 76] [3, 17, 22, 25, 65, 67, 90] [2, 4, 18, 23, 65, 66, 95] [1, 5, 19, 27, 67, 79, 92] [20, 28, 31, 39, 56, 68, 70] [3, 21, 32, 40, 68, 69, 71] [10, 22, 30, 33, 49, 70, 93] [5, 11, 23, 34, 70, 71, 83] [24, 29, 31, 35, 59, 72, 74] [7, 25, 30, 32, 72, 73, 75] [3, 26, 33, 37, 50, 74, 94] [4, 9, 27, 34, 74, 75, 86] [28, 35, 38, 43, 48, 76, 78] [29, 36, 39, 42, 51, 64, 77] [17, 30, 37, 40, 52, 77, 78] [13, 31, 36, 38, 66, 79, 81] [10, 14, 32, 37, 79, 80, 82] [7, 15, 33, 40, 54, 81, 95] [8, 12, 16, 34, 81, 82, 90] [13, 35, 42, 46, 49, 83, 85] [24, 36, 41, 43, 55, 71, 84] [15, 25, 37, 44, 56, 84, 85] [20, 38, 41, 42, 50, 86, 88] [13, 39, 43, 45, 58, 75, 87] [14, 22, 40, 44, 59, 87, 88] [31, 41, 45, 46, 53, 62, 89] [24, 28, 42, 45, 54, 90, 92] [20, 29, 43, 46, 65, 82, 91] [21, 26, 30, 44, 66, 91, 92] [29, 38, 45, 47, 57, 69, 93] [28, 36, 46, 47, 60, 73, 94] [35, 39, 41, 47, 67, 80, 95]

Code ID 96-20-6 · download JSON · raw on GitHub