[[72,26,4]] · QEC Challenge

[[72,26,4]] d =

kd²/n

5.778

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)

[20, 30, 56, 66]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[51, 53, 54, 62]

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)×168 (3,4)×1224 (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): 168 (2,6): 672 (3,4): 1224 (3,6): 2112 (3,8): 8352 (3,10): 384

trapping sets H_Z (1,4)×72 (2,4)×168 (3,4)×1224 (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): 168 (2,6): 672 (3,4): 1224 (3,6): 2112 (3,8): 8352 (3,10): 384

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_k26.txt, line 3. nonidentity supports a=[2, 4, 20], b=[4, 6, 16].

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

[[72,26,4]] — 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 = 5.78.

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_k26.txt, line 3. This row is SmallGroup(36,3) with nonidentity GAP supports a=[2, 4, 20], b=[4, 6, 16]. 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=26), 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(36,3), 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, 20], b=[4, 6, 16] 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) = 26. 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, 32, 36, 39, 52, 70] [0, 1, 8, 12, 37, 43, 59, 71] [2, 5, 10, 15, 37, 38, 46, 55] [0, 3, 31, 35, 36, 39, 63, 64] [4, 12, 23, 32, 40, 48, 52, 70] [1, 5, 7, 26, 36, 41, 48, 50] [2, 6, 18, 22, 41, 42, 53, 62] [3, 4, 7, 19, 37, 43, 67, 68] [1, 3, 4, 8, 44, 55, 59, 71] [9, 13, 20, 25, 42, 45, 56, 65] [2, 10, 14, 26, 38, 43, 44, 46] [5, 11, 15, 22, 37, 47, 55, 58] [4, 12, 31, 35, 40, 48, 63, 64] [6, 13, 17, 33, 38, 49, 58, 60] [1, 7, 14, 15, 39, 40, 41, 50] [8, 14, 15, 19, 36, 48, 51, 62] [9, 16, 28, 30, 49, 52, 63, 69] [10, 11, 17, 29, 42, 50, 51, 53] [6, 10, 11, 18, 41, 54, 62, 65] [0, 7, 12, 19, 44, 55, 67, 68] [9, 20, 24, 33, 45, 53, 54, 56] [13, 21, 25, 30, 42, 57, 65, 66] [11, 14, 22, 26, 43, 44, 47, 58] [16, 23, 27, 35, 45, 59, 66, 67] [6, 17, 24, 25, 46, 47, 49, 60] [18, 24, 25, 29, 38, 58, 61, 69] [5, 8, 19, 26, 39, 40, 51, 62] [20, 21, 27, 34, 52, 60, 61, 63] [16, 20, 21, 28, 49, 64, 69, 70] [2, 17, 22, 29, 50, 51, 54, 65] [21, 24, 30, 33, 53, 54, 57, 66] [16, 27, 31, 32, 56, 57, 59, 67] [28, 31, 32, 34, 45, 66, 68, 71] [13, 18, 29, 33, 46, 47, 61, 69] [9, 27, 30, 34, 60, 61, 64, 70] [23, 28, 34, 35, 56, 57, 68, 71]

H_Z (36 checks, sparse supports)

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

Code ID 72-26-4 · download JSON · raw on GitHub