← back to the board
[[144,16,7]] d ≤
n
144
k
16
d
7
kd²/n
5.444
w
6
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · 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 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[27, 46, 98, 99, 113, 118, 142]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[14, 31, 32, 33, 43, 86, 103]
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)×144 (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): 144 (3,4): 2160 (3,6): 6912 (3,8): 3240 (3,10): 288
trapping sets H_Z (1,2)×72 (2,2)×72 (3,2)×144 (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): 144 (3,4): 2160 (3,6): 6912 (3,8): 3240 (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

[[144,16,7]] — two-block group-algebra code on SmallGroup(72,1)

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 = 5.44.

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_k16.txt, line 4. This row is SmallGroup(72,1) with nonidentity GAP supports a=[5], b=[2, 16, 46]. 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=16), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 7, not an exact certificate. The published distance (d=7) 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(72,1), 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] and b=[2, 16, 46] 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) = 16. 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, 34, 72, 88, 90, 139] [1, 8, 72, 73, 96, 107] [2, 49, 73, 74, 101, 117] [3, 51, 75, 78, 104, 130] [0, 4, 76, 106, 128, 138] [5, 53, 77, 107, 127, 128] [6, 19, 74, 78, 111, 122] [7, 21, 75, 79, 114, 124] [8, 23, 80, 89, 106, 116] [9, 24, 81, 88, 89, 117] [10, 62, 79, 82, 119, 132] [2, 11, 83, 97, 116, 121] [12, 64, 84, 96, 97, 122] [3, 13, 85, 112, 123, 129] [14, 65, 86, 111, 112, 124] [4, 15, 72, 87, 126, 127] [5, 16, 88, 109, 125, 126] [15, 17, 76, 89, 108, 109] [18, 36, 82, 90, 127, 135] [19, 38, 91, 102, 121, 129] [20, 39, 92, 101, 102, 130] [21, 41, 93, 105, 123, 131] [22, 42, 94, 104, 105, 132] [23, 25, 73, 87, 88, 95] [24, 44, 77, 87, 96, 133] [25, 45, 76, 77, 80, 97] [10, 26, 98, 115, 131, 134] [27, 70, 99, 114, 115, 135] [11, 28, 74, 95, 96, 100] [12, 29, 81, 95, 101, 136] [28, 30, 80, 81, 83, 102] [13, 31, 75, 103, 110, 111] [14, 32, 92, 104, 110, 137] [31, 33, 85, 91, 92, 105] [16, 34, 77, 106, 108, 143] [17, 35, 87, 90, 107, 143] [36, 54, 108, 120, 134, 138] [37, 55, 109, 119, 120, 139] [38, 40, 78, 100, 101, 110] [39, 57, 84, 100, 111, 140] [40, 58, 83, 84, 91, 112] [41, 43, 79, 103, 104, 113] [42, 59, 86, 103, 114, 141] [43, 60, 85, 86, 93, 115] [1, 44, 76, 81, 116, 125] [45, 61, 72, 95, 117, 125] [26, 46, 82, 113, 114, 118] [27, 47, 94, 113, 119, 142] [46, 48, 93, 94, 98, 120] [29, 49, 80, 84, 121, 133] [30, 50, 73, 100, 122, 133] [32, 51, 86, 91, 123, 140] [33, 52, 78, 103, 124, 140] [35, 53, 89, 125, 138, 139] [54, 56, 90, 118, 119, 126] [55, 66, 99, 118, 127, 143] [56, 67, 98, 99, 108, 128] [6, 57, 83, 92, 129, 136] [58, 68, 74, 110, 130, 136] [7, 59, 85, 94, 131, 137] [60, 69, 75, 113, 132, 137] [9, 61, 97, 106, 107, 133] [47, 62, 93, 99, 134, 141] [48, 63, 79, 118, 135, 141] [50, 64, 102, 116, 117, 136] [52, 65, 105, 129, 130, 137] [18, 66, 98, 109, 138, 142] [67, 71, 82, 126, 139, 142] [20, 68, 112, 121, 122, 140] [22, 69, 115, 123, 124, 141] [63, 70, 120, 131, 132, 142] [37, 71, 128, 134, 135, 143]
H_Z (72 checks, sparse supports)
[0, 1, 15, 45, 72, 76] [1, 2, 23, 50, 73, 116] [2, 6, 28, 58, 74, 83] [3, 7, 31, 60, 75, 85] [4, 17, 25, 44, 76, 87] [5, 24, 25, 34, 77, 88] [3, 6, 38, 52, 78, 129] [7, 10, 41, 63, 79, 131] [8, 25, 30, 49, 73, 80] [9, 29, 30, 44, 81, 133] [10, 18, 46, 67, 82, 98] [11, 30, 40, 57, 83, 100] [12, 39, 40, 49, 84, 101] [13, 33, 43, 59, 85, 103] [14, 42, 43, 51, 86, 104] [15, 23, 24, 35, 87, 89] [0, 9, 16, 23, 88, 106] [8, 9, 17, 53, 89, 107] [0, 18, 35, 54, 90, 138] [19, 33, 40, 51, 78, 91] [20, 32, 33, 57, 92, 140] [21, 43, 48, 62, 79, 93] [22, 47, 48, 59, 94, 141] [23, 28, 29, 45, 80, 95] [1, 12, 24, 28, 81, 96] [11, 12, 25, 61, 95, 97] [26, 48, 56, 66, 98, 118] [27, 55, 56, 62, 99, 119] [28, 38, 39, 50, 100, 102] [2, 20, 29, 38, 101, 121] [19, 20, 30, 64, 102, 122] [31, 41, 42, 52, 103, 105] [3, 22, 32, 41, 104, 123] [21, 22, 33, 65, 105, 124] [4, 8, 34, 61, 72, 106] [1, 5, 35, 61, 107, 125] [17, 34, 36, 56, 90, 108] [16, 17, 37, 66, 109, 143] [31, 32, 38, 58, 91, 110] [6, 14, 31, 39, 92, 111] [13, 14, 40, 68, 110, 112] [41, 46, 47, 60, 93, 113] [7, 27, 42, 46, 94, 114] [26, 27, 43, 69, 113, 115] [8, 11, 44, 64, 96, 116] [2, 9, 45, 64, 97, 117] [46, 54, 55, 63, 118, 120] [10, 37, 47, 54, 119, 134] [36, 37, 48, 70, 120, 135] [11, 19, 49, 68, 74, 121] [6, 12, 50, 68, 122, 136] [13, 21, 51, 69, 75, 123] [7, 14, 52, 69, 124, 137] [16, 44, 45, 53, 77, 125] [15, 16, 54, 67, 108, 126] [5, 15, 18, 55, 109, 127] [4, 5, 56, 71, 126, 128] [13, 19, 57, 65, 111, 129] [3, 20, 58, 65, 112, 130] [21, 26, 59, 70, 114, 131] [10, 22, 60, 70, 115, 132] [24, 49, 50, 61, 117, 133] [26, 36, 62, 71, 82, 134] [18, 27, 63, 71, 135, 142] [29, 57, 58, 64, 84, 136] [32, 59, 60, 65, 86, 137] [4, 36, 53, 66, 127, 138] [0, 37, 53, 67, 128, 139] [39, 51, 52, 68, 130, 140] [42, 62, 63, 69, 132, 141] [47, 66, 67, 70, 99, 142] [34, 35, 55, 71, 139, 143]
Code ID 144-16-7 · download JSON · raw on GitHub