[[72,4,10]] · QEC Challenge

[[72,4,10]] d =

kd²/n

5.556

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · 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 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[3, 6, 12, 13, 19, 31, 37, 41, 61, 64]

d_Z 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[12, 20, 22, 25, 28, 33, 46, 51, 61, 64]

certificate exact, d = 10 · CryptoMiniSat 5.14 SAT

X: no logical < 10 exists; Z: no logical < 10 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): 3348 (3,8): 1620 (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): 3348 (3,8): 1620 (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

[[72,4,10]] — 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 = 5.56.

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_wt6wtL2_order36_k4.txt, line 1. This row is SmallGroup(36,2) with nonidentity GAP supports a=[10], b=[2, 3, 17]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 10, not an exact certificate. The published distance (d=10) 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=[10] and b=[2, 3, 17] 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) = 4. 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, 34, 36, 42, 53, 65] [1, 35, 36, 37, 58, 68] [2, 3, 36, 38, 50, 62] [3, 29, 37, 39, 63, 70] [4, 18, 37, 40, 44, 52] [5, 6, 37, 38, 41, 66] [6, 32, 39, 42, 48, 71] [7, 23, 39, 40, 43, 49] [8, 9, 38, 44, 59, 68] [0, 9, 39, 41, 45, 69] [10, 11, 40, 41, 46, 61] [8, 11, 42, 43, 47, 54] [12, 28, 43, 48, 55, 63] [13, 14, 41, 44, 49, 70] [1, 14, 42, 45, 50, 57] [15, 16, 43, 45, 46, 51] [13, 16, 36, 47, 52, 59] [17, 31, 47, 48, 53, 60] [2, 18, 45, 49, 54, 71] [19, 20, 46, 49, 55, 67] [4, 20, 47, 50, 51, 56] [21, 22, 48, 51, 57, 69] [19, 22, 52, 53, 58, 64] [5, 23, 50, 54, 59, 65] [24, 25, 51, 54, 55, 60] [7, 25, 38, 52, 56, 61] [26, 27, 53, 56, 57, 62] [24, 27, 40, 58, 63, 67] [10, 28, 56, 59, 60, 64] [29, 30, 57, 60, 65, 71] [12, 30, 58, 61, 62, 66] [15, 31, 44, 61, 64, 67] [32, 33, 62, 64, 65, 68] [17, 33, 46, 63, 66, 69] [21, 34, 66, 67, 68, 70] [26, 35, 55, 69, 70, 71]

H_Z (36 checks, sparse supports)

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

Code ID 72-4-10 · download JSON · raw on GitHub