← back to the board
[[144,4,16]] d ≤
n
144
k
4
d
16
kd²/n
7.111
w
6
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[6, 14, 23, 29, 42, 69, 72, 73, 74, 85, 88, 94, 129, 131, 137, 142]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[1, 9, 13, 14, 15, 16, 26, 30, 42, 43, 101, 106, 117, 123, 124, 130]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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)×72 (2,2)×72 (3,2)×72 (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 (1,4): 72 (2,2): 72 (2,4): 576 (2,6): 432 (3,2): 72 (3,4): 2304 (3,6): 6768 (3,8): 4320 (3,10): 288
trapping sets H_Z (1,2)×72 (2,2)×72 (3,2)×72 (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 (1,4): 72 (2,2): 72 (2,4): 576 (2,6): 432 (3,2): 72 (3,4): 2304 (3,6): 6768 (3,8): 4320 (3,10): 288

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

[[144,4,16]] — two-block group-algebra code on SmallGroup(72,10)

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 = 7.11.

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_wt6wtL2_order72_k4.txt, line 48. This row is SmallGroup(72,10) with nonidentity GAP supports a=[15], b=[17, 19, 43]. 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 ≤ 16, not an exact certificate. The published distance (d=16) 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(72,10), 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=[15] and b=[17, 19, 43] 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) = 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 72 (max weight 6) · Z-checks 72 (max weight 6)
H_X (72 checks, sparse supports)
[0, 65, 72, 95, 107, 143] [1, 69, 73, 87, 117, 142] [2, 70, 74, 122, 128, 132] [3, 4, 75, 91, 114, 125] [4, 52, 76, 89, 115, 139] [5, 31, 76, 77, 116, 126] [6, 71, 78, 120, 124, 130] [7, 8, 79, 83, 104, 133] [8, 60, 80, 97, 105, 135] [9, 41, 80, 81, 106, 118] [10, 11, 73, 82, 136, 138] [11, 63, 83, 102, 110, 141] [12, 46, 83, 84, 93, 140] [13, 14, 85, 97, 109, 137] [0, 14, 78, 86, 106, 131] [15, 16, 86, 87, 111, 132] [13, 16, 72, 88, 108, 133] [17, 51, 79, 88, 89, 138] [18, 19, 72, 90, 134, 140] [19, 67, 91, 100, 112, 137] [20, 54, 85, 91, 92, 136] [21, 22, 89, 93, 99, 141] [1, 22, 74, 94, 116, 123] [23, 24, 94, 95, 101, 124] [21, 24, 73, 96, 98, 125] [25, 59, 75, 96, 97, 134] [26, 27, 79, 98, 130, 142] [2, 27, 80, 99, 121, 127] [28, 29, 81, 99, 100, 143] [26, 29, 74, 101, 113, 129] [30, 62, 101, 102, 109, 114] [3, 31, 90, 103, 117, 124] [32, 33, 73, 103, 104, 128] [5, 33, 75, 92, 105, 141] [34, 35, 93, 105, 106, 130] [32, 35, 77, 94, 107, 127] [36, 37, 75, 108, 122, 143] [6, 37, 76, 109, 119, 129] [38, 39, 77, 109, 110, 142] [36, 39, 78, 103, 111, 121] [40, 66, 99, 104, 111, 112] [7, 41, 82, 107, 113, 132] [42, 43, 72, 113, 114, 120] [9, 43, 79, 84, 115, 137] [44, 45, 85, 115, 116, 122] [42, 45, 81, 86, 117, 119] [10, 46, 94, 112, 118, 135] [47, 48, 91, 95, 118, 119] [12, 48, 82, 96, 120, 139] [49, 50, 97, 120, 121, 126] [47, 50, 84, 90, 122, 131] [15, 51, 80, 85, 110, 123] [52, 53, 81, 123, 124, 140] [17, 53, 87, 112, 113, 125] [18, 54, 86, 102, 126, 139] [55, 56, 83, 87, 126, 127] [20, 56, 88, 90, 128, 135] [57, 58, 89, 118, 128, 129] [55, 58, 82, 92, 123, 130] [23, 59, 76, 93, 100, 131] [60, 61, 77, 131, 132, 136] [25, 61, 95, 102, 103, 133] [28, 62, 78, 98, 115, 134] [63, 64, 111, 116, 134, 135] [30, 64, 100, 108, 117, 136] [34, 65, 96, 104, 129, 137] [38, 66, 74, 105, 108, 138] [67, 68, 101, 106, 138, 139] [40, 68, 98, 107, 110, 140] [44, 69, 88, 114, 121, 141] [49, 70, 92, 119, 133, 142] [57, 71, 84, 125, 127, 143]
H_Z (72 checks, sparse supports)
[0, 16, 18, 42, 72, 86] [1, 10, 24, 32, 73, 94] [2, 22, 29, 66, 74, 99] [3, 25, 33, 36, 75, 103] [4, 5, 37, 59, 75, 76] [5, 35, 38, 60, 77, 105] [6, 14, 39, 62, 78, 109] [7, 17, 26, 43, 79, 113] [8, 9, 27, 51, 79, 80] [9, 28, 45, 52, 81, 115] [10, 41, 48, 58, 82, 118] [7, 11, 12, 55, 82, 83] [12, 43, 50, 71, 84, 120] [13, 20, 44, 51, 85, 88] [14, 15, 45, 54, 85, 86] [1, 15, 53, 55, 87, 123] [16, 17, 56, 69, 87, 88] [4, 17, 21, 57, 89, 125] [18, 31, 50, 56, 90, 126] [3, 19, 20, 47, 90, 91] [20, 33, 58, 70, 92, 128] [12, 21, 34, 59, 93, 96] [22, 23, 35, 46, 93, 94] [0, 23, 47, 61, 95, 131] [24, 25, 48, 65, 95, 96] [8, 13, 25, 49, 97, 133] [24, 26, 62, 68, 98, 101] [21, 27, 28, 40, 98, 99] [19, 28, 59, 64, 100, 134] [23, 29, 30, 67, 100, 101] [11, 30, 54, 61, 102, 136] [31, 32, 39, 61, 77, 103] [7, 32, 40, 65, 104, 107] [8, 33, 34, 66, 104, 105] [9, 14, 34, 67, 106, 137] [0, 35, 41, 68, 106, 107] [16, 36, 64, 66, 108, 111] [13, 30, 37, 38, 108, 109] [11, 38, 51, 68, 110, 138] [15, 39, 40, 63, 110, 111] [19, 40, 46, 53, 112, 140] [29, 41, 42, 53, 81, 113] [3, 30, 42, 69, 114, 117] [4, 43, 44, 62, 114, 115] [5, 22, 44, 63, 116, 141] [1, 31, 45, 64, 116, 117] [9, 46, 47, 57, 84, 118] [37, 45, 47, 70, 119, 122] [6, 42, 48, 49, 119, 120] [27, 39, 49, 69, 121, 142] [2, 36, 44, 50, 121, 122] [22, 51, 52, 58, 89, 123] [6, 23, 31, 52, 76, 124] [3, 24, 53, 71, 124, 125] [5, 49, 54, 55, 92, 126] [27, 35, 55, 71, 127, 130] [2, 32, 56, 57, 127, 128] [29, 37, 57, 65, 129, 143] [6, 26, 34, 58, 129, 130] [14, 50, 59, 60, 97, 131] [2, 15, 41, 60, 80, 132] [7, 16, 61, 70, 132, 133] [18, 25, 62, 63, 102, 134] [8, 46, 56, 63, 83, 135] [10, 20, 60, 64, 135, 136] [13, 19, 43, 65, 72, 137] [10, 17, 66, 67, 112, 138] [4, 48, 54, 67, 91, 139] [12, 18, 52, 68, 139, 140] [11, 21, 33, 69, 73, 141] [1, 26, 38, 70, 74, 142] [0, 28, 36, 71, 78, 143]
Code ID 144-4-16 · download JSON · raw on GitHub