[[72,28,2]] · QEC Challenge

[[72,28,2]] d =

kd²/n

1.556

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[56, 64]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[67, 71]

certificate exact, d = 2 · scipy/HiGHS MILP

X: no logical < 2 exists; Z: no logical < 2 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)×36 (2,0)×54 (3,3)×228 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 36 (1,4): 36 (2,0): 54 (2,4): 72 (2,5): 432 (3,3): 228 (3,4): 648 (3,5): 864 (3,6): 1944 (3,7): 2160

trapping sets H_Z (1,3)×36 (2,0)×54 (3,3)×228 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 36 (1,4): 36 (2,0): 54 (2,4): 72 (2,5): 432 (3,3): 228 (3,4): 648 (3,5): 864 (3,6): 1944 (3,7): 2160

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(36,8), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL3_order36_k28.txt, line 1. nonidentity supports a=[3, 9], b=[5, 7, 18].

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

[[72,28,2]] — two-block group-algebra code on SmallGroup(36,8)

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 = 1.56.

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_order36_k28.txt, line 1. This row is SmallGroup(36,8) with nonidentity GAP supports a=[3, 9], b=[5, 7, 18]. 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=28), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, not an exact certificate. The published distance (d=2) 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(36,8), 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, 9], b=[5, 7, 18] 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 = 72 - rank(HX) - rank(HZ) = 28. 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 36 (max weight 7) · Z-checks 36 (max weight 7)

H_X (36 checks, sparse supports)

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

H_Z (36 checks, sparse supports)

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

Code ID 72-28-2 · download JSON · raw on GitHub