[[98,14,6]] · QEC Challenge

[[98,14,6]] d =

kd²/n

5.143

X/Z

Share this result

Distance

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

[45, 80, 86, 90, 91, 94]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[16, 23, 29, 41, 79, 94]

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 6 · H_Z 6

qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×49 (2,2)×49 (3,2)×98 (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 (1,4): 49 (2,2): 49 (2,4): 392 (2,6): 294 (3,2): 98 (3,4): 1666 (3,6): 4116 (3,8): 2205 (3,10): 196

trapping sets H_Z (1,2)×49 (2,2)×49 (3,2)×98 (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 (1,4): 49 (2,2): 49 (2,4): 392 (2,6): 294 (3,2): 98 (3,4): 1666 (3,6): 4116 (3,8): 2205 (3,10): 196

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 ≤ 6 (computed)

How this code was found

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

[[98,14,6]] — two-block group-algebra code on SmallGroup(49,1)

Direction & hypothesis

Target: unrestricted weight-6 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 = 5.14.

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_wt6wtL2_order49_k14.txt, line 1. This row is SmallGroup(49,1) with nonidentity GAP supports a=[3], b=[2, 6, 14]. 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=14), 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(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] and b=[2, 6, 14] 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) = 14. 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 6) · Z-checks 49 (max weight 6)

H_X (49 checks, sparse supports)

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

H_Z (49 checks, sparse supports)

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

Code ID 98-14-6 · download JSON · raw on GitHub