[[72,6,9]] · QEC Challenge

[[72,6,9]] d =

kd²/n

6.75

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · w_X = 6, w_Z = 6 (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)

[12, 14, 15, 26, 30, 31, 61, 65, 67]

d_Z 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[22, 26, 33, 41, 55, 56, 61, 62, 69]

certificate exact, d = 9 · scipy/HiGHS MILP

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 6

qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×36 (2,2)×36 (3,2)×36 (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 (1,4): 36 (2,2): 36 (2,4): 288 (2,6): 216 (3,2): 36 (3,4): 1224 (3,6): 3312 (3,8): 1728 (3,10): 144

trapping sets H_Z (1,2)×36 (2,2)×36 (3,2)×36 (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 (1,4): 36 (2,2): 36 (2,4): 288 (2,6): 216 (3,2): 36 (3,4): 1224 (3,6): 3312 (3,8): 1728 (3,10): 144

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-18

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 ≤ 6 (computed)

How this code was found

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

[[72,6,9]] — two-block group-algebra code on SmallGroup(36,2)

Direction & hypothesis

Target: unrestricted weight-6 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 = 6.75.

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_wt6wtL2_order36_k6.txt, line 1. This row is SmallGroup(36,2) with nonidentity GAP supports a=[3], b=[2, 10, 28]. 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

Union Alpha (maker currently anonymous), 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=[3] and b=[2, 10, 28] 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) = 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 36 (max weight 6) · Z-checks 36 (max weight 6)

H_X (36 checks, sparse supports)

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

H_Z (36 checks, sparse supports)

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

Code ID 72-6-9 · download JSON · raw on GitHub