[[70,22,4]] · QEC Challenge

[[70,22,4]] d =

kd²/n

5.029

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[17, 25, 40, 44]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[25, 29, 44, 52]

certificate exact, d = 4 · scipy/HiGHS cutoff IP

X: no logical < 4 exists; Z: no logical < 4 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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×70 (2,4)×140 (3,4)×560 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 70 (2,4): 140 (2,6): 700 (3,4): 560 (3,6): 2730 (3,8): 9730 (3,10): 560

trapping sets H_Z (1,4)×70 (2,4)×140 (3,4)×560 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 70 (2,4): 140 (2,6): 700 (3,4): 560 (3,6): 2730 (3,8): 9730 (3,10): 560

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_wt8wtL4_order35_k22.txt, line 1. nonidentity supports a=[2, 3, 5], b=[4, 6, 13].

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

[[70,22,4]] — two-block group-algebra code on SmallGroup(35,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 5.03.

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_wt8wtL4_order35_k22.txt, line 1. This row is SmallGroup(35,1) with nonidentity GAP supports a=[2, 3, 5], b=[4, 6, 13]. 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=22), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 4, not an exact certificate. The published distance 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=[2, 3, 5], b=[4, 6, 13] 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) = 22. 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 8) · Z-checks 35 (max weight 8)

H_X (35 checks, sparse supports)

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

H_Z (35 checks, sparse supports)

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

Code ID 70-22-4 · download JSON · raw on GitHub