[[72,24,2]] · QEC Challenge

[[72,24,2]] d =

kd²/n

1.333

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)

[51, 61]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[26, 33]

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 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)

trapping sets H_X (1,2)×36 (2,0)×18 (3,2)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 36 (1,5): 36 (2,0): 18 (2,3): 72 (2,4): 36 (2,5): 216 (2,6): 18 (2,8): 216 (3,2): 144 (3,3): 432 (3,4): 108 (3,5): 864 (3,6): 648 (3,7): 108 (3,8): 2844 (3,9): 228 (3,11): 1836

trapping sets H_Z (1,2)×36 (2,0)×18 (3,2)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 36 (1,5): 36 (2,0): 18 (2,3): 72 (2,4): 36 (2,5): 216 (2,6): 18 (2,8): 216 (3,2): 144 (3,3): 432 (3,4): 108 (3,5): 864 (3,6): 648 (3,7): 108 (3,8): 2844 (3,9): 228 (3,11): 1836

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(36,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt7wtL2_order36_k24.txt, line 1. nonidentity supports a=[4], b=[4, 6, 16, 27].

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

[[72,24,2]] — two-block group-algebra code on SmallGroup(36,2)

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 = 1.33.

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_wt7wtL2_order36_k24.txt, line 1. This row is SmallGroup(36,2) with nonidentity GAP supports a=[4], b=[4, 6, 16, 27]. 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=24), 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(36,2), 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=[4], b=[4, 6, 16, 27] 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) = 24. 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, 3, 36, 39, 59, 67, 71] [1, 6, 37, 42, 44, 55, 65] [2, 9, 38, 42, 45, 52, 63] [0, 3, 36, 39, 49, 60, 68] [4, 11, 40, 47, 59, 67, 71] [5, 14, 36, 40, 41, 48, 50] [1, 6, 37, 42, 54, 64, 70] [7, 16, 43, 44, 52, 55, 65] [8, 18, 44, 50, 54, 61, 69] [2, 9, 37, 38, 43, 45, 53] [10, 20, 42, 46, 52, 56, 63] [4, 11, 40, 47, 49, 60, 68] [12, 22, 48, 58, 59, 67, 71] [13, 23, 38, 46, 49, 57, 59] [5, 14, 39, 41, 47, 50, 58] [15, 25, 36, 40, 48, 51, 61] [7, 16, 43, 52, 54, 64, 70] [17, 27, 44, 53, 55, 63, 65] [8, 18, 41, 44, 51, 54, 62] [19, 28, 50, 55, 61, 64, 69] [10, 20, 37, 43, 46, 53, 56] [21, 30, 42, 52, 57, 63, 66] [12, 22, 48, 49, 58, 60, 68] [13, 23, 45, 49, 56, 59, 66] [24, 31, 38, 46, 57, 60, 67] [15, 25, 39, 47, 51, 58, 61] [26, 33, 36, 40, 48, 62, 69] [17, 27, 53, 54, 63, 64, 70] [19, 28, 41, 51, 55, 62, 64] [29, 34, 50, 61, 65, 69, 70] [21, 30, 37, 43, 53, 57, 66] [24, 31, 45, 56, 60, 66, 67] [32, 35, 38, 46, 57, 68, 71] [26, 33, 39, 47, 58, 62, 69] [29, 34, 41, 51, 62, 65, 70] [32, 35, 45, 56, 66, 68, 71]

H_Z (36 checks, sparse supports)

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

Code ID 72-24-2 · download JSON · raw on GitHub