[[144,48,2]] · QEC Challenge

[[144,48,2]] d =

144

kd²/n

1.333

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 7, w_Z = 7 (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)

[93, 113]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[1, 8]

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 7 · H_Z 7

qubit degrees H_X 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)

trapping sets H_X (1,2)×72 (2,0)×36 (3,2)×288 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,2): 72 (1,5): 72 (2,0): 36 (2,3): 144 (2,4): 72 (2,5): 432 (2,6): 36 (2,8): 432 (3,2): 288 (3,3): 864 (3,4): 216 (3,5): 1296 (3,6): 1296 (3,7): 1080 (3,8): 5688 (3,9): 456 (3,11): 4104

trapping sets H_Z (1,2)×72 (2,0)×36 (3,2)×288 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,2): 72 (1,5): 72 (2,0): 36 (2,3): 144 (2,4): 72 (2,5): 432 (2,6): 36 (2,8): 432 (3,2): 288 (3,3): 864 (3,4): 216 (3,5): 1296 (3,6): 1296 (3,7): 1080 (3,8): 5688 (3,9): 456 (3,11): 4104

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(72,12), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt7wtL2_order72_k48.txt, line 1. nonidentity supports a=[5], b=[2, 5, 22, 58].

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

[[144,48,2]] — two-block group-algebra code on SmallGroup(72,12)

Direction & hypothesis

Target: unrestricted weight-8 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 = 1.33.

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_wt7wtL2_order72_k48.txt, line 1. This row is SmallGroup(72,12) with nonidentity GAP supports a=[5], b=[2, 5, 22, 58]. 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=48), 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 (d=2) 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(72,12), 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=[5], b=[2, 5, 22, 58] 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 = 144 - rank(HX) - rank(HZ) = 48. 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 72 (max weight 7) · Z-checks 72 (max weight 7)

H_X (72 checks, sparse supports)

[0, 4, 72, 76, 94, 138, 141] [1, 8, 72, 73, 80, 100, 104] [2, 12, 74, 84, 111, 115, 143] [3, 14, 73, 75, 86, 110, 114] [0, 4, 72, 76, 79, 127, 131] [5, 16, 77, 88, 126, 130, 133] [6, 20, 74, 77, 78, 92, 121] [7, 22, 75, 79, 94, 119, 123] [1, 8, 73, 76, 80, 120, 124] [9, 24, 81, 82, 85, 89, 96] [10, 27, 82, 99, 126, 130, 133] [11, 29, 78, 81, 83, 101, 129] [2, 12, 74, 84, 91, 95, 139] [13, 31, 85, 94, 103, 138, 141] [3, 14, 75, 80, 86, 128, 132] [15, 33, 87, 90, 93, 97, 105] [5, 16, 77, 88, 108, 112, 116] [17, 35, 89, 107, 111, 115, 143] [18, 37, 82, 85, 89, 90, 109] [19, 39, 83, 87, 91, 111, 135] [6, 20, 78, 84, 88, 92, 136] [21, 41, 72, 93, 100, 104, 113] [7, 22, 79, 86, 94, 134, 137] [23, 43, 95, 98, 102, 106, 115] [9, 24, 81, 96, 99, 103, 107] [25, 45, 74, 77, 97, 117, 121] [26, 46, 90, 93, 97, 98, 118] [10, 27, 82, 99, 108, 112, 116] [28, 48, 100, 111, 115, 120, 143] [11, 29, 83, 92, 96, 101, 140] [30, 50, 73, 102, 110, 114, 122] [13, 31, 79, 85, 103, 127, 131] [32, 52, 104, 124, 126, 130, 133] [15, 33, 87, 105, 109, 113, 117] [34, 53, 78, 81, 106, 125, 129] [17, 35, 89, 91, 95, 107, 139] [36, 54, 98, 102, 106, 108, 126] [18, 37, 90, 99, 103, 107, 109] [38, 56, 74, 77, 110, 121, 128] [19, 39, 91, 101, 105, 111, 142] [40, 58, 75, 112, 119, 123, 130] [21, 41, 76, 93, 113, 120, 124] [42, 60, 82, 85, 89, 114, 132] [23, 43, 95, 115, 118, 122, 125] [44, 61, 83, 87, 116, 133, 135] [25, 45, 84, 88, 97, 117, 136] [26, 46, 98, 109, 113, 117, 118] [47, 62, 78, 81, 119, 129, 134] [28, 48, 91, 95, 100, 120, 139] [49, 64, 94, 121, 136, 138, 141] [30, 50, 80, 102, 122, 128, 132] [51, 65, 90, 93, 97, 123, 137] [32, 52, 104, 108, 112, 116, 124] [34, 53, 92, 96, 106, 125, 140] [36, 54, 108, 118, 122, 125, 126] [55, 66, 83, 87, 127, 135, 138] [38, 56, 84, 88, 110, 128, 136] [57, 68, 72, 100, 104, 129, 140] [40, 58, 86, 112, 130, 134, 137] [59, 69, 98, 102, 106, 131, 141] [42, 60, 99, 103, 107, 114, 132] [44, 61, 101, 105, 116, 133, 142] [47, 62, 92, 96, 119, 134, 140] [63, 70, 73, 110, 114, 135, 142] [49, 64, 79, 121, 127, 131, 136] [51, 65, 109, 113, 117, 123, 137] [55, 66, 101, 105, 127, 138, 142] [67, 71, 75, 119, 123, 139, 143] [57, 68, 76, 120, 124, 129, 140] [59, 69, 118, 122, 125, 131, 141] [63, 70, 80, 128, 132, 135, 142] [67, 71, 86, 134, 137, 139, 143]

H_Z (72 checks, sparse supports)

[0, 1, 4, 21, 57, 72, 76] [1, 3, 8, 30, 63, 73, 80] [2, 6, 12, 25, 38, 74, 84] [3, 7, 14, 40, 67, 75, 86] [0, 4, 8, 41, 68, 72, 76] [5, 6, 16, 25, 38, 77, 88] [6, 11, 20, 34, 47, 78, 92] [4, 7, 22, 31, 64, 79, 94] [1, 8, 14, 50, 70, 73, 80] [9, 11, 24, 34, 47, 81, 96] [9, 10, 18, 27, 42, 82, 99] [11, 19, 29, 44, 55, 83, 101] [2, 12, 20, 45, 56, 74, 84] [9, 13, 18, 31, 42, 85, 103] [3, 14, 22, 58, 71, 75, 86] [15, 19, 33, 44, 55, 87, 105] [5, 16, 20, 45, 56, 77, 88] [9, 17, 18, 35, 42, 89, 107] [15, 18, 26, 37, 51, 90, 109] [12, 19, 35, 39, 48, 91, 111] [6, 20, 29, 53, 62, 78, 92] [15, 21, 26, 41, 51, 93, 113] [0, 7, 13, 22, 49, 79, 94] [12, 23, 35, 43, 48, 95, 115] [9, 24, 29, 53, 62, 81, 96] [15, 25, 26, 45, 51, 97, 117] [23, 26, 36, 46, 59, 98, 118] [10, 24, 27, 37, 60, 82, 99] [1, 21, 28, 48, 57, 100, 120] [11, 29, 39, 61, 66, 83, 101] [23, 30, 36, 50, 59, 102, 122] [13, 24, 31, 37, 60, 85, 103] [1, 21, 32, 52, 57, 104, 124] [15, 33, 39, 61, 66, 87, 105] [23, 34, 36, 53, 59, 106, 125] [17, 24, 35, 37, 60, 89, 107] [16, 27, 36, 52, 54, 108, 126] [18, 33, 37, 46, 65, 90, 109] [3, 30, 38, 56, 63, 110, 128] [2, 17, 19, 28, 39, 91, 111] [16, 27, 40, 52, 58, 112, 130] [21, 33, 41, 46, 65, 93, 113] [3, 30, 42, 60, 63, 114, 132] [2, 17, 23, 28, 43, 95, 115] [16, 27, 44, 52, 61, 116, 133] [25, 33, 45, 46, 65, 97, 117] [26, 43, 46, 54, 69, 98, 118] [7, 40, 47, 62, 67, 119, 134] [8, 28, 41, 48, 68, 100, 120] [6, 25, 38, 49, 64, 121, 136] [30, 43, 50, 54, 69, 102, 122] [7, 40, 51, 65, 67, 123, 137] [8, 32, 41, 52, 68, 104, 124] [34, 43, 53, 54, 69, 106, 125] [5, 10, 32, 36, 54, 108, 126] [4, 31, 55, 64, 66, 127, 138] [14, 38, 50, 56, 70, 110, 128] [11, 34, 47, 57, 68, 129, 140] [5, 10, 32, 40, 58, 112, 130] [4, 31, 59, 64, 69, 131, 141] [14, 42, 50, 60, 70, 114, 132] [5, 10, 32, 44, 61, 116, 133] [22, 47, 58, 62, 71, 119, 134] [19, 44, 55, 63, 70, 135, 142] [20, 45, 49, 56, 64, 121, 136] [22, 51, 58, 65, 71, 123, 137] [0, 13, 49, 55, 66, 127, 138] [12, 35, 48, 67, 71, 139, 143] [29, 53, 57, 62, 68, 129, 140] [0, 13, 49, 59, 69, 131, 141] [39, 61, 63, 66, 70, 135, 142] [2, 17, 28, 67, 71, 139, 143]

Code ID 144-48-2 · download JSON · raw on GitHub