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[[144,36,4]] d =
n
144
k
36
d
4
kd²/n
4.0
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · 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 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[69, 132, 135, 140]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[40, 58, 119, 134]
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 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)×144 (3,2)×648 (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): 144 (2,4): 468 (2,6): 360 (3,2): 648 (3,4): 2160 (3,6): 3312 (3,8): 1584 (3,10): 144
trapping sets H_Z (1,2)×72 (2,2)×144 (3,2)×648 (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): 144 (2,4): 468 (2,6): 360 (3,2): 648 (3,4): 2160 (3,6): 3312 (3,8): 1584 (3,10): 144

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

[[144,36,4]] — 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 = 4.00.

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_k36.txt, line 9. This row is SmallGroup(72,10) with nonidentity GAP supports a=[7], b=[5, 8, 11]. 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=36), 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 (d=4) 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=[7] and b=[5, 8, 11] 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) = 36. 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, 19, 72, 76, 132, 135] [1, 2, 73, 80, 124, 139] [2, 8, 74, 83, 124, 143] [3, 37, 73, 74, 75, 86] [4, 6, 72, 76, 141, 142] [5, 39, 77, 88, 93, 98] [0, 6, 78, 91, 132, 142] [7, 10, 72, 78, 79, 94] [8, 11, 73, 80, 137, 143] [9, 12, 81, 85, 96, 108] [10, 22, 72, 82, 91, 99] [1, 11, 74, 83, 137, 139] [12, 24, 84, 85, 101, 126] [13, 54, 79, 82, 85, 103] [14, 18, 75, 80, 83, 86] [15, 56, 81, 84, 87, 105] [16, 20, 77, 88, 113, 118] [17, 58, 89, 107, 114, 119] [3, 18, 73, 83, 90, 109] [4, 19, 78, 91, 135, 141] [5, 20, 92, 93, 111, 118] [21, 26, 75, 90, 93, 113] [22, 27, 76, 79, 91, 94] [23, 28, 77, 92, 95, 115] [24, 29, 81, 96, 103, 126] [25, 30, 97, 104, 117, 127] [26, 41, 75, 98, 109, 118] [7, 27, 76, 78, 82, 99] [28, 43, 77, 100, 111, 120] [9, 29, 84, 101, 103, 108] [30, 45, 102, 104, 122, 138] [31, 36, 85, 94, 99, 103] [32, 66, 95, 100, 104, 123] [33, 38, 87, 96, 101, 105] [34, 68, 97, 102, 106, 125] [35, 40, 89, 107, 131, 134] [13, 36, 79, 99, 108, 126] [14, 37, 74, 80, 90, 109] [15, 38, 81, 101, 110, 128] [16, 39, 92, 98, 111, 113] [17, 40, 112, 114, 130, 134] [41, 46, 86, 93, 109, 113] [42, 47, 87, 110, 114, 131] [43, 48, 88, 95, 111, 115] [44, 49, 89, 112, 116, 133] [45, 50, 97, 117, 123, 138] [21, 46, 86, 90, 98, 118] [47, 59, 87, 119, 128, 134] [23, 48, 88, 92, 100, 120] [49, 61, 89, 121, 130, 136] [25, 50, 102, 122, 123, 127] [51, 55, 104, 115, 120, 123] [52, 71, 116, 121, 124, 137] [53, 57, 106, 117, 122, 125] [31, 54, 82, 94, 108, 126] [32, 55, 95, 120, 127, 138] [33, 56, 84, 96, 110, 128] [34, 57, 97, 122, 129, 140] [35, 58, 112, 119, 130, 131] [59, 62, 105, 114, 128, 131] [60, 63, 106, 129, 132, 141] [61, 64, 107, 116, 130, 133] [42, 62, 105, 110, 119, 134] [63, 69, 106, 135, 140, 142] [44, 64, 107, 112, 121, 136] [65, 67, 124, 133, 136, 137] [51, 66, 100, 115, 127, 138] [52, 67, 116, 136, 139, 143] [53, 68, 102, 117, 129, 140] [69, 70, 125, 132, 140, 141] [60, 70, 125, 129, 135, 142] [65, 71, 121, 133, 139, 143]
H_Z (72 checks, sparse supports)
[0, 4, 7, 10, 72, 78] [1, 3, 8, 18, 73, 83] [2, 3, 11, 37, 73, 74] [3, 14, 21, 26, 75, 90] [0, 4, 22, 27, 76, 91] [5, 16, 23, 28, 77, 92] [6, 7, 19, 27, 76, 78] [7, 13, 22, 36, 79, 99] [1, 8, 14, 37, 74, 80] [9, 15, 24, 38, 81, 101] [10, 13, 27, 54, 79, 82] [2, 11, 14, 18, 80, 83] [12, 15, 29, 56, 81, 84] [9, 12, 13, 31, 85, 108] [3, 14, 41, 46, 86, 109] [15, 33, 42, 47, 87, 110] [5, 16, 43, 48, 88, 111] [17, 35, 44, 49, 89, 112] [18, 21, 37, 46, 86, 90] [6, 10, 19, 22, 72, 91] [20, 23, 39, 48, 88, 92] [5, 20, 21, 41, 93, 118] [7, 22, 31, 54, 82, 94] [23, 32, 43, 55, 95, 120] [9, 24, 33, 56, 84, 96] [25, 34, 45, 57, 97, 122] [5, 26, 39, 46, 93, 98] [10, 27, 31, 36, 94, 99] [28, 32, 48, 66, 95, 100] [12, 29, 33, 38, 96, 101] [30, 34, 50, 68, 97, 102] [13, 24, 29, 31, 103, 126] [25, 30, 32, 51, 104, 127] [15, 33, 59, 62, 105, 128] [34, 53, 60, 63, 106, 129] [17, 35, 61, 64, 107, 130] [9, 29, 36, 54, 103, 108] [18, 26, 37, 41, 75, 109] [38, 42, 56, 62, 105, 110] [20, 28, 39, 43, 77, 111] [40, 44, 58, 64, 107, 112] [16, 21, 39, 41, 98, 113] [17, 40, 42, 59, 114, 134] [23, 43, 51, 66, 100, 115] [44, 52, 61, 67, 116, 136] [25, 45, 53, 68, 102, 117] [16, 20, 26, 46, 113, 118] [17, 47, 58, 62, 114, 119] [28, 48, 51, 55, 115, 120] [49, 52, 64, 71, 116, 121] [30, 50, 53, 57, 117, 122] [32, 45, 50, 51, 123, 138] [1, 2, 52, 65, 124, 139] [34, 53, 69, 70, 125, 140] [12, 24, 36, 54, 85, 126] [25, 50, 55, 66, 123, 127] [38, 47, 56, 59, 87, 128] [57, 60, 68, 70, 125, 129] [40, 49, 58, 61, 89, 130] [35, 42, 58, 59, 119, 131] [0, 6, 60, 69, 132, 142] [44, 61, 65, 71, 121, 133] [35, 40, 47, 62, 131, 134] [0, 19, 63, 70, 132, 135] [49, 64, 65, 67, 133, 136] [8, 11, 52, 65, 137, 143] [30, 45, 55, 66, 104, 138] [1, 11, 67, 71, 137, 139] [57, 63, 68, 69, 106, 140] [4, 19, 60, 69, 135, 141] [4, 6, 63, 70, 141, 142] [2, 8, 67, 71, 124, 143]
Code ID 144-36-4 · download JSON · raw on GitHub