[[98,6,7]] · QEC Challenge

[[98,6,7]] d =

kd²/n

3.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 7, d_Z = 7 · 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 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[50, 53, 57, 62, 68, 75, 82]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[3, 7, 12, 18, 25, 32, 38]

certificate exact, d = 7 · scipy/HiGHS MILP

X: no logical < 7 exists; Z: no logical < 7 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)×49 (2,2)×49 (3,2)×49 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 49 (1,3): 49 (2,2): 49 (2,3): 294 (2,4): 147 (3,2): 49 (3,3): 931 (3,4): 1617 (3,5): 588 (3,6): 294 (3,7): 49

trapping sets H_Z (1,2)×49 (2,2)×49 (3,2)×49 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 49 (1,3): 49 (2,2): 49 (2,3): 294 (2,4): 147 (3,2): 49 (3,3): 931 (3,4): 1617 (3,5): 588 (3,6): 294 (3,7): 49

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(49,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt5wtL2_order49_k6.txt, line 1. nonidentity supports a=[3], b=[2, 12].

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

[[98,6,7]] — two-block group-algebra code on SmallGroup(49,1)

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 = 3.00.

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_wt5wtL2_order49_k6.txt, line 1. This row is SmallGroup(49,1) with nonidentity GAP supports a=[3], b=[2, 12]. 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 ≤ 7, not an exact certificate. The published distance (d=7) 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(49,1), 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=[3], b=[2, 12] 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 = 98 - 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 49 (max weight 5) · Z-checks 49 (max weight 5)

H_X (49 checks, sparse supports)

[0, 27, 49, 90, 97] [1, 33, 49, 50, 93] [0, 2, 51, 70, 94] [3, 38, 50, 52, 95] [1, 4, 51, 53, 96] [2, 5, 54, 59, 77] [6, 42, 52, 55, 76] [3, 7, 53, 56, 97] [4, 8, 54, 57, 64] [5, 9, 58, 65, 83] [10, 45, 55, 59, 82] [6, 11, 49, 56, 60] [7, 12, 57, 61, 70] [8, 13, 58, 62, 71] [9, 14, 63, 72, 88] [15, 47, 59, 64, 87] [10, 16, 50, 60, 65] [11, 17, 51, 61, 66] [12, 18, 62, 67, 77] [13, 19, 63, 68, 78] [14, 20, 69, 79, 92] [21, 48, 64, 70, 91] [15, 22, 52, 65, 71] [16, 23, 53, 66, 72] [17, 24, 54, 67, 73] [18, 25, 68, 74, 83] [19, 26, 69, 75, 84] [20, 27, 76, 85, 95] [21, 28, 55, 71, 77] [22, 29, 56, 72, 78] [23, 30, 57, 73, 79] [24, 31, 58, 74, 80] [25, 32, 75, 81, 88] [26, 33, 76, 82, 89] [28, 34, 60, 78, 83] [29, 35, 61, 79, 84] [30, 36, 62, 80, 85] [31, 37, 63, 81, 86] [32, 38, 82, 87, 92] [34, 39, 66, 84, 88] [35, 40, 67, 85, 89] [36, 41, 68, 86, 90] [37, 42, 69, 87, 91] [39, 43, 73, 89, 92] [40, 44, 74, 90, 93] [41, 45, 75, 91, 94] [43, 46, 80, 93, 95] [44, 47, 81, 94, 96] [46, 48, 86, 96, 97]

H_Z (49 checks, sparse supports)

[0, 1, 11, 49, 51] [1, 3, 16, 50, 53] [2, 4, 17, 51, 54] [3, 6, 22, 52, 56] [4, 7, 23, 53, 57] [5, 8, 24, 54, 58] [6, 10, 28, 55, 60] [7, 11, 29, 56, 61] [8, 12, 30, 57, 62] [9, 13, 31, 58, 63] [5, 10, 15, 59, 65] [11, 16, 34, 60, 66] [12, 17, 35, 61, 67] [13, 18, 36, 62, 68] [14, 19, 37, 63, 69] [8, 15, 21, 64, 71] [9, 16, 22, 65, 72] [17, 23, 39, 66, 73] [18, 24, 40, 67, 74] [19, 25, 41, 68, 75] [20, 26, 42, 69, 76] [2, 12, 21, 70, 77] [13, 22, 28, 71, 78] [14, 23, 29, 72, 79] [24, 30, 43, 73, 80] [25, 31, 44, 74, 81] [26, 32, 45, 75, 82] [6, 27, 33, 49, 76] [5, 18, 28, 77, 83] [19, 29, 34, 78, 84] [20, 30, 35, 79, 85] [31, 36, 46, 80, 86] [32, 37, 47, 81, 87] [10, 33, 38, 50, 82] [9, 25, 34, 83, 88] [26, 35, 39, 84, 89] [27, 36, 40, 85, 90] [37, 41, 48, 86, 91] [15, 38, 42, 52, 87] [14, 32, 39, 88, 92] [33, 40, 43, 89, 93] [0, 41, 44, 90, 94] [21, 42, 45, 55, 91] [20, 38, 43, 92, 95] [1, 44, 46, 93, 96] [2, 45, 47, 59, 94] [3, 27, 46, 95, 97] [4, 47, 48, 64, 96] [0, 7, 48, 70, 97]

Code ID 98-6-7 · download JSON · raw on GitHub