[[78,26,3]] · QEC Challenge

[[78,26,3]] d =

kd²/n

3.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[20, 22, 24]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[13, 16, 52]

certificate exact, d = 3 · scipy/HiGHS MILP

X: no logical < 3 exists; Z: no logical < 3 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 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)×39 (2,2)×117 (3,0)×65 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 39 (1,4): 39 (2,2): 117 (2,4): 234 (2,6): 78 (3,0): 65 (3,2): 390 (3,4): 975 (3,6): 897 (3,8): 156

trapping sets H_Z (1,2)×39 (2,2)×117 (3,0)×65 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 39 (1,4): 39 (2,2): 117 (2,4): 234 (2,6): 78 (3,0): 65 (3,2): 390 (3,4): 975 (3,6): 897 (3,8): 156

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(39,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order39_k26.txt, line 1. nonidentity supports a=[2], 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 ≤ 6 (computed)

How this code was found

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

[[78,26,3]] — two-block group-algebra code on SmallGroup(39,2)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 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_wt6wtL2_order39_k26.txt, line 1. This row is SmallGroup(39,2) with nonidentity GAP supports a=[2], 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=26), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 3, not an exact certificate. The published distance (d=3) 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(39,2), 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=[2], 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 = 78 - rank(HX) - rank(HZ) = 26. 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 39 (max weight 6) · Z-checks 39 (max weight 6)

H_X (39 checks, sparse supports)

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

H_Z (39 checks, sparse supports)

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

Code ID 78-26-3 · download JSON · raw on GitHub