[[72,22,4]] · QEC Challenge

[[72,22,4]] d =

kd²/n

4.889

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · 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 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[10, 28, 56, 64]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[27, 31, 53, 67]

certificate exact, d = 4 · scipy/HiGHS MILP

X: no logical < 4 exists; Z: no logical < 4 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 (2,4)×360 (3,4)×1368 (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 (2,4): 360 (2,6): 288 (3,4): 1368 (3,6): 3456 (3,8): 1872

trapping sets H_Z (1,4)×72 (2,4)×360 (3,4)×1368 (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 (2,4): 360 (2,6): 288 (3,4): 1368 (3,6): 3456 (3,8): 1872

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_k22.txt, line 1. nonidentity supports a=[2, 3, 6], b=[2, 10, 15].

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

[[72,22,4]] — 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 = 4.89.

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_k22.txt, line 1. This row is SmallGroup(36,2) with nonidentity GAP supports a=[2, 3, 6], b=[2, 10, 15]. 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=22), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 4, 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, 3, 6], b=[2, 10, 15] 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) = 22. 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, 6, 29, 35, 36, 42, 68, 70] [0, 1, 29, 32, 36, 37, 70, 71] [0, 2, 6, 14, 37, 38, 39, 50] [1, 3, 32, 34, 37, 39, 65, 71] [4, 8, 16, 23, 40, 49, 52, 54] [0, 1, 2, 5, 38, 39, 41, 42] [3, 6, 34, 35, 39, 42, 65, 68] [4, 7, 8, 13, 40, 43, 54, 59] [2, 8, 14, 23, 41, 44, 45, 59] [1, 3, 5, 9, 36, 41, 42, 45] [4, 10, 16, 25, 43, 46, 47, 61] [7, 11, 13, 18, 43, 44, 47, 59] [12, 19, 27, 31, 48, 60, 63, 64] [2, 5, 8, 13, 44, 45, 49, 50] [3, 6, 9, 14, 36, 37, 45, 50] [4, 7, 10, 15, 46, 47, 51, 52] [11, 16, 18, 23, 44, 47, 49, 52] [12, 17, 19, 24, 48, 53, 64, 67] [5, 9, 13, 18, 38, 49, 50, 54] [10, 19, 25, 31, 51, 55, 56, 67] [7, 11, 15, 20, 40, 51, 52, 56] [12, 21, 27, 33, 53, 57, 58, 69] [17, 22, 24, 28, 53, 55, 58, 67] [9, 14, 18, 23, 38, 41, 54, 59] [10, 15, 19, 24, 55, 56, 60, 61] [11, 16, 20, 25, 40, 43, 56, 61] [12, 17, 21, 26, 57, 58, 62, 63] [22, 27, 28, 31, 55, 58, 60, 63] [15, 20, 24, 28, 46, 60, 61, 64] [21, 29, 33, 35, 62, 65, 66, 71] [17, 22, 26, 30, 48, 62, 63, 66] [20, 25, 28, 31, 46, 51, 64, 67] [21, 26, 29, 32, 65, 66, 68, 69] [22, 27, 30, 33, 48, 53, 66, 69] [26, 30, 32, 34, 57, 68, 69, 70] [30, 33, 34, 35, 57, 62, 70, 71]

H_Z (36 checks, sparse supports)

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

Code ID 72-22-4 · download JSON · raw on GitHub