[[70,6,9]] · QEC Challenge

[[70,6,9]] d =

kd²/n

6.943

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · 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 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[7, 15, 17, 20, 24, 29, 31, 45, 54]

d_Z 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[10, 16, 39, 46, 53, 60, 63, 67, 69]

certificate exact, d = 9 · CryptoMiniSat 5.14 SAT

X: no logical < 9 exists; Z: no logical < 9 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 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)×35 (2,3)×35 (3,3)×35 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 35 (1,4): 35 (2,3): 35 (2,4): 105 (2,5): 350 (2,6): 210 (3,3): 35 (3,4): 385 (3,5): 945 (3,6): 2765 (3,7): 3710 (3,8): 1435 (3,9): 420 (3,10): 140

trapping sets H_Z (1,3)×35 (2,3)×35 (3,3)×35 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 35 (1,4): 35 (2,3): 35 (2,4): 105 (2,5): 350 (2,6): 210 (3,3): 35 (3,4): 385 (3,5): 945 (3,6): 2765 (3,7): 3710 (3,8): 1435 (3,9): 420 (3,10): 140

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(35,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order35_k6.txt, line 1. nonidentity supports a=[3, 14], b=[3, 13, 20].

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 ≤ 8 (computed)

How this code was found

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

[[70,6,9]] — two-block group-algebra code on SmallGroup(35,1)

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 = 6.94.

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_order35_k6.txt, line 1. This row is SmallGroup(35,1) with nonidentity GAP supports a=[3, 14], b=[3, 13, 20]. 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 ≤ 9, not an exact certificate. The published distance (d=9) 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(35,1), 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, 14], b=[3, 13, 20] 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 = 70 - 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 35 (max weight 7) · Z-checks 35 (max weight 7)

H_X (35 checks, sparse supports)

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

H_Z (35 checks, sparse supports)

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

Code ID 70-6-9 · download JSON · raw on GitHub