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[[144,54,2]] d =
n
144
k
54
d
2
kd²/n
1.5
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[85, 93]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[49, 57]
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 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 (2,0)×72 (3,0)×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 (2,0): 72 (2,2): 144 (2,4): 288 (2,6): 288 (3,0): 144 (3,2): 216 (3,4): 1944 (3,6): 2592 (3,8): 1728
trapping sets H_Z (1,2)×72 (2,0)×36 (3,2)×1152 (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 (2,0): 36 (2,2): 144 (2,4): 504 (3,2): 1152 (3,4): 1728 (3,6): 1080 (3,8): 144

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(72,10), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order72_k54.txt, line 1. nonidentity supports a=[2], b=[2, 11, 19].
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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[144,54,2]] — two-block group-algebra code on SmallGroup(72,10)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.50.

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_k54.txt, line 1. This row is SmallGroup(72,10) with nonidentity GAP supports a=[2], b=[2, 11, 19]. 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=54), 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(72,10), 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=[2], b=[2, 11, 19] 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) = 54. 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, 1, 72, 73, 135, 143] [0, 1, 72, 73, 139, 142] [2, 6, 74, 91, 124, 132] [3, 7, 74, 75, 79, 91] [4, 8, 76, 80, 139, 142] [5, 9, 77, 81, 98, 126] [2, 6, 78, 83, 124, 132] [3, 7, 75, 78, 79, 83] [4, 8, 76, 80, 135, 143] [5, 9, 77, 81, 108, 118] [10, 18, 72, 73, 82, 109] [11, 19, 78, 83, 137, 141] [12, 20, 84, 85, 93, 111] [13, 21, 82, 85, 93, 109] [14, 22, 78, 83, 86, 94] [15, 23, 84, 87, 95, 111] [16, 24, 88, 96, 108, 118] [17, 25, 89, 97, 119, 138] [10, 18, 72, 73, 90, 99] [11, 19, 74, 91, 137, 141] [12, 20, 85, 92, 93, 101] [13, 21, 85, 90, 93, 99] [14, 22, 74, 86, 91, 94] [15, 23, 87, 92, 95, 101] [16, 24, 88, 96, 98, 126] [17, 25, 89, 97, 127, 134] [26, 36, 75, 79, 98, 126] [27, 37, 76, 80, 90, 99] [28, 38, 77, 81, 100, 128] [29, 39, 92, 101, 103, 113] [30, 40, 102, 104, 114, 130] [31, 41, 90, 99, 103, 113] [32, 42, 100, 104, 114, 128] [33, 43, 92, 101, 105, 115] [34, 44, 102, 106, 116, 130] [35, 45, 107, 117, 127, 134] [26, 36, 75, 79, 108, 118] [27, 37, 76, 80, 82, 109] [28, 38, 77, 81, 110, 120] [29, 39, 84, 103, 111, 113] [30, 40, 104, 112, 114, 122] [31, 41, 82, 103, 109, 113] [32, 42, 104, 110, 114, 120] [33, 43, 84, 105, 111, 115] [34, 44, 106, 112, 116, 122] [35, 45, 107, 117, 119, 138] [46, 54, 86, 94, 108, 118] [47, 55, 87, 95, 119, 138] [48, 56, 88, 96, 110, 120] [49, 57, 89, 97, 121, 140] [50, 58, 112, 122, 123, 131] [51, 59, 110, 120, 123, 131] [52, 60, 121, 124, 132, 140] [53, 61, 112, 122, 125, 133] [46, 54, 86, 94, 98, 126] [47, 55, 87, 95, 127, 134] [48, 56, 88, 96, 100, 128] [49, 57, 89, 97, 129, 136] [50, 58, 102, 123, 130, 131] [51, 59, 100, 123, 128, 131] [52, 60, 124, 129, 132, 136] [53, 61, 102, 125, 130, 133] [62, 66, 105, 115, 127, 134] [63, 67, 106, 116, 135, 143] [64, 68, 107, 117, 129, 136] [65, 69, 129, 136, 137, 141] [62, 66, 105, 115, 119, 138] [63, 67, 106, 116, 139, 142] [64, 68, 107, 117, 121, 140] [65, 69, 121, 137, 140, 141] [70, 71, 125, 133, 139, 142] [70, 71, 125, 133, 135, 143]
H_Z (72 checks, sparse supports)
[0, 1, 10, 18, 72, 73] [0, 1, 10, 18, 72, 73] [2, 3, 19, 22, 74, 78] [3, 7, 26, 36, 75, 79] [4, 8, 27, 37, 76, 80] [5, 9, 28, 38, 77, 81] [6, 7, 11, 14, 74, 78] [3, 7, 26, 36, 75, 79] [4, 8, 27, 37, 76, 80] [5, 9, 28, 38, 77, 81] [10, 13, 37, 41, 82, 90] [6, 7, 11, 14, 83, 91] [12, 15, 39, 43, 84, 92] [12, 13, 20, 21, 85, 93] [14, 22, 46, 54, 86, 94] [15, 23, 47, 55, 87, 95] [16, 24, 48, 56, 88, 96] [17, 25, 49, 57, 89, 97] [18, 21, 27, 31, 82, 90] [2, 3, 19, 22, 83, 91] [20, 23, 29, 33, 84, 92] [12, 13, 20, 21, 85, 93] [14, 22, 46, 54, 86, 94] [15, 23, 47, 55, 87, 95] [16, 24, 48, 56, 88, 96] [17, 25, 49, 57, 89, 97] [5, 24, 26, 54, 98, 108] [18, 21, 27, 31, 99, 109] [28, 32, 56, 59, 100, 110] [20, 23, 29, 33, 101, 111] [30, 34, 58, 61, 102, 112] [29, 31, 39, 41, 103, 113] [30, 32, 40, 42, 104, 114] [33, 43, 62, 66, 105, 115] [34, 44, 63, 67, 106, 116] [35, 45, 64, 68, 107, 117] [9, 16, 36, 46, 98, 108] [10, 13, 37, 41, 99, 109] [38, 42, 48, 51, 100, 110] [12, 15, 39, 43, 101, 111] [40, 44, 50, 53, 102, 112] [29, 31, 39, 41, 103, 113] [30, 32, 40, 42, 104, 114] [33, 43, 62, 66, 105, 115] [34, 44, 63, 67, 106, 116] [35, 45, 64, 68, 107, 117] [9, 16, 36, 46, 118, 126] [17, 45, 47, 66, 119, 127] [38, 42, 48, 51, 120, 128] [49, 52, 68, 69, 121, 129] [40, 44, 50, 53, 122, 130] [50, 51, 58, 59, 123, 131] [2, 6, 52, 60, 124, 132] [53, 61, 70, 71, 125, 133] [5, 24, 26, 54, 118, 126] [25, 35, 55, 62, 119, 127] [28, 32, 56, 59, 120, 128] [57, 60, 64, 65, 121, 129] [30, 34, 58, 61, 122, 130] [50, 51, 58, 59, 123, 131] [2, 6, 52, 60, 124, 132] [53, 61, 70, 71, 125, 133] [25, 35, 55, 62, 134, 138] [0, 8, 63, 71, 135, 139] [57, 60, 64, 65, 136, 140] [11, 19, 65, 69, 137, 141] [17, 45, 47, 66, 134, 138] [1, 4, 67, 70, 135, 139] [49, 52, 68, 69, 136, 140] [11, 19, 65, 69, 137, 141] [1, 4, 67, 70, 142, 143] [0, 8, 63, 71, 142, 143]
Code ID 144-54-2 · download JSON · raw on GitHub