[[96,6,6]] · QEC Challenge

[[96,6,6]] d =

kd²/n

2.25

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 5, w_Z = 5 (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)

[29, 40, 70, 71, 76, 77]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[14, 21, 27, 34, 38, 42]

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 5 · H_Z 5

qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)

trapping sets H_X (1,2)×48 (2,2)×48 (3,1)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 48 (1,3): 48 (2,2): 48 (2,3): 288 (2,4): 144 (3,1): 24 (3,2): 48 (3,3): 824 (3,4): 1584 (3,5): 624 (3,6): 288 (3,7): 48

trapping sets H_Z (1,2)×48 (2,2)×48 (3,1)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 48 (1,3): 48 (2,2): 48 (2,3): 288 (2,4): 144 (3,1): 24 (3,2): 48 (3,3): 824 (3,4): 1584 (3,5): 624 (3,6): 288 (3,7): 48

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(48,32), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt5wtL2_order48_k6.txt, line 6. nonidentity supports a=[7], b=[3, 25].

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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[96,6,6]] — two-block group-algebra code on SmallGroup(48,32)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-5 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 2.25.

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_wt5wtL2_order48_k6.txt, line 6. This row is SmallGroup(48,32) with nonidentity GAP supports a=[7], b=[3, 25]. 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 ≤ 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

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(48,32), 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=[7], b=[3, 25] 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) = 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 48 (max weight 5) · Z-checks 48 (max weight 5)

H_X (48 checks, sparse supports)

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

H_Z (48 checks, sparse supports)

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

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