[[72,38,2]] · QEC Challenge

[[72,38,2]] d =

kd²/n

2.111

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)

[51, 56]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[21, 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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×72 (2,0)×72 (3,4)×2556 (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): 360 (3,4): 2556 (3,8): 288

trapping sets H_Z (1,4)×72 (2,0)×72 (3,4)×2556 (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): 72 (2,4): 360 (3,4): 2556 (3,8): 288

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_wt8wtL4_order36_k38.txt, line 1. nonidentity supports a=[2, 4, 7], b=[3, 4, 10].

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

[[72,38,2]] — two-block group-algebra code on SmallGroup(36,2)

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 = 2.11.

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_wt8wtL4_order36_k38.txt, line 1. This row is SmallGroup(36,2) with nonidentity GAP supports a=[2, 4, 7], b=[3, 4, 10]. 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=38), 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(36,2), 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=[2, 4, 7], b=[3, 4, 10] 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) = 38. 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, 1, 3, 6, 36, 39, 65, 70] [0, 1, 3, 6, 37, 42, 68, 71] [2, 5, 9, 14, 36, 38, 39, 45] [0, 1, 3, 6, 36, 39, 65, 70] [4, 7, 11, 16, 40, 44, 47, 54] [2, 5, 9, 14, 37, 41, 42, 50] [0, 1, 3, 6, 37, 42, 68, 71] [4, 7, 11, 16, 43, 49, 52, 59] [8, 13, 18, 23, 38, 44, 45, 54] [2, 5, 9, 14, 36, 38, 39, 45] [10, 15, 20, 25, 40, 46, 47, 56] [4, 7, 11, 16, 40, 44, 47, 54] [12, 17, 22, 27, 48, 55, 58, 64] [8, 13, 18, 23, 41, 49, 50, 59] [2, 5, 9, 14, 37, 41, 42, 50] [10, 15, 20, 25, 43, 51, 52, 61] [4, 7, 11, 16, 43, 49, 52, 59] [12, 17, 22, 27, 53, 60, 63, 67] [8, 13, 18, 23, 38, 44, 45, 54] [19, 24, 28, 31, 46, 55, 56, 64] [10, 15, 20, 25, 40, 46, 47, 56] [21, 26, 30, 33, 48, 57, 58, 66] [12, 17, 22, 27, 48, 55, 58, 64] [8, 13, 18, 23, 41, 49, 50, 59] [19, 24, 28, 31, 51, 60, 61, 67] [10, 15, 20, 25, 43, 51, 52, 61] [21, 26, 30, 33, 53, 62, 63, 69] [12, 17, 22, 27, 53, 60, 63, 67] [19, 24, 28, 31, 46, 55, 56, 64] [29, 32, 34, 35, 57, 65, 66, 70] [21, 26, 30, 33, 48, 57, 58, 66] [19, 24, 28, 31, 51, 60, 61, 67] [29, 32, 34, 35, 62, 68, 69, 71] [21, 26, 30, 33, 53, 62, 63, 69] [29, 32, 34, 35, 57, 65, 66, 70] [29, 32, 34, 35, 62, 68, 69, 71]

H_Z (36 checks, sparse supports)

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

Code ID 72-38-2 · download JSON · raw on GitHub