[[72,37,2]] · QEC Challenge

[[72,37,2]] d =

kd²/n

2.056

X/Z

Share this result

Distance

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

witness operator (support, 2 qubits)

[13, 25]

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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×72 (2,0)×72 (3,4)×1620 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 72 (2,0): 72 (2,4): 288 (2,6): 144 (3,4): 1620 (3,6): 1728 (3,8): 864

trapping sets H_Z (1,4)×72 (2,0)×54 (3,0)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 72 (2,0): 54 (2,4): 396 (3,0): 144 (3,4): 2376 (3,8): 504

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(36,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order36_k37.txt, line 1. nonidentity supports a=[2, 4, 8], b=[4, 5, 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

[[72,37,2]] — two-block group-algebra code on SmallGroup(36,3)

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 = 2.06.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order36_k37.txt, line 1. This row is SmallGroup(36,3) with nonidentity GAP supports a=[2, 4, 8], b=[4, 5, 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=37), 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 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(36,3), 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, 4, 8], b=[4, 5, 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 = 72 - rank(HX) - rank(HZ) = 37. 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 8) · Z-checks 36 (max weight 8)

H_X (36 checks, sparse supports)

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

H_Z (36 checks, sparse supports)

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

Code ID 72-37-2 · download JSON · raw on GitHub