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[[162,54,3]] d =
n
162
k
54
d
3
kd²/n
3.0
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[95, 111, 130]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[6, 19, 39]
certificate exact, d = 3 · scipy/HiGHS MILP
X: no logical < 3 exists; Z: no logical < 3 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 8 · H_Z 8
qubit degrees H_X 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×81 (2,2)×81 (3,0)×54 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 81 (1,6): 81 (2,2): 81 (2,4): 243 (2,6): 567 (2,8): 243 (2,10): 486 (3,0): 54 (3,2): 243 (3,4): 1458 (3,6): 2943 (3,8): 5103 (3,10): 7614 (3,12): 3051 (3,14): 3159
trapping sets H_Z (1,2)×81 (2,2)×81 (3,0)×54 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 81 (1,6): 81 (2,2): 81 (2,4): 243 (2,6): 567 (2,8): 243 (2,10): 486 (3,0): 54 (3,2): 243 (3,4): 1458 (3,6): 2943 (3,8): 5103 (3,10): 7614 (3,12): 3051 (3,14): 3159

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(81,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order81_k54.txt, line 1. nonidentity supports a=[4], b=[2, 3, 4, 8, 11].
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

[[162,54,3]] — two-block group-algebra code on SmallGroup(81,3)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 3.00.

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_wt8wtL2_order81_k54.txt, line 1. This row is SmallGroup(81,3) with nonidentity GAP supports a=[4], b=[2, 3, 4, 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=54), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 3, not an exact certificate. The published distance 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(81,3), 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=[4], b=[2, 3, 4, 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 = 162 - 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 81 (max weight 8) · Z-checks 81 (max weight 8)
H_X (81 checks, sparse supports)
[0, 12, 81, 90, 93, 117, 125, 152] [1, 21, 81, 82, 93, 99, 102, 138] [2, 26, 81, 83, 93, 107, 151, 159] [0, 3, 81, 84, 90, 105, 117, 137] [4, 29, 85, 86, 106, 110, 115, 144] [5, 34, 82, 86, 102, 112, 115, 147] [6, 39, 82, 87, 91, 102, 107, 120] [1, 7, 81, 82, 84, 88, 99, 118] [8, 42, 85, 89, 110, 119, 123, 153] [9, 44, 83, 90, 107, 125, 149, 158] [2, 10, 81, 83, 84, 91, 135, 159] [11, 47, 85, 92, 110, 113, 128, 133] [3, 12, 84, 93, 105, 125, 137, 152] [4, 13, 85, 86, 94, 97, 106, 126] [14, 49, 95, 98, 127, 130, 136, 156] [15, 52, 86, 96, 100, 115, 120, 133] [5, 16, 82, 86, 88, 97, 112, 131] [17, 55, 89, 98, 123, 132, 136, 157] [18, 57, 87, 90, 99, 105, 120, 138] [6, 19, 82, 83, 87, 88, 100, 107] [20, 60, 89, 101, 108, 123, 128, 141] [7, 21, 84, 88, 93, 102, 118, 138] [8, 22, 85, 89, 94, 103, 119, 139] [23, 62, 95, 104, 130, 140, 143, 160] [9, 24, 83, 90, 91, 105, 158, 161] [25, 63, 92, 106, 112, 128, 131, 144] [10, 26, 84, 91, 93, 107, 135, 151] [11, 27, 85, 92, 94, 96, 108, 133] [28, 65, 95, 109, 130, 134, 146, 150] [13, 29, 94, 97, 110, 115, 126, 144] [14, 30, 95, 98, 111, 116, 127, 145] [31, 66, 96, 99, 112, 118, 133, 147] [15, 32, 86, 87, 96, 97, 113, 120] [33, 69, 98, 114, 121, 136, 141, 150] [16, 34, 88, 97, 102, 115, 131, 147] [17, 35, 89, 98, 103, 116, 132, 148] [36, 71, 104, 117, 143, 149, 152, 161] [18, 37, 87, 99, 100, 105, 118, 125] [38, 72, 101, 106, 119, 126, 141, 153] [19, 39, 83, 88, 91, 100, 102, 120] [20, 40, 89, 92, 101, 103, 121, 128] [41, 74, 104, 122, 129, 143, 146, 155] [22, 42, 94, 103, 110, 123, 139, 153] [23, 43, 95, 104, 111, 124, 140, 154] [24, 44, 91, 105, 107, 125, 149, 161] [25, 45, 92, 106, 108, 126, 131, 147] [46, 75, 109, 127, 132, 146, 148, 156] [27, 47, 94, 96, 108, 110, 113, 128] [28, 48, 95, 109, 111, 114, 129, 150] [30, 49, 111, 116, 130, 136, 145, 156] [31, 50, 96, 112, 113, 118, 131, 138] [51, 76, 114, 119, 132, 139, 150, 157] [32, 52, 87, 97, 100, 113, 115, 133] [33, 53, 98, 101, 114, 116, 134, 141] [54, 78, 117, 135, 142, 152, 155, 159] [35, 55, 103, 116, 123, 136, 148, 157] [36, 56, 104, 117, 124, 137, 149, 158] [37, 57, 90, 100, 118, 120, 125, 138] [38, 58, 101, 119, 121, 126, 139, 144] [59, 79, 122, 127, 140, 145, 155, 160] [40, 60, 92, 103, 108, 121, 123, 141] [41, 61, 104, 109, 122, 124, 142, 146] [43, 62, 111, 124, 130, 143, 154, 160] [45, 63, 108, 112, 126, 128, 144, 147] [46, 64, 109, 127, 129, 145, 148, 157] [48, 65, 111, 114, 129, 130, 134, 146] [50, 66, 99, 113, 131, 133, 138, 147] [51, 67, 114, 132, 134, 139, 148, 153] [68, 80, 135, 140, 149, 154, 159, 161] [53, 69, 101, 116, 121, 134, 136, 150] [54, 70, 117, 122, 135, 137, 151, 155] [56, 71, 124, 137, 143, 152, 158, 161] [58, 72, 106, 121, 139, 141, 144, 153] [59, 73, 122, 140, 142, 145, 154, 156] [61, 74, 109, 124, 129, 142, 143, 155] [64, 75, 129, 132, 145, 146, 156, 157] [67, 76, 119, 134, 148, 150, 153, 157] [68, 77, 135, 149, 151, 154, 158, 160] [70, 78, 122, 137, 142, 151, 152, 159] [73, 79, 127, 142, 154, 155, 156, 160] [77, 80, 140, 151, 158, 159, 160, 161]
H_Z (81 checks, sparse supports)
[0, 1, 2, 3, 7, 10, 81, 84] [1, 5, 6, 7, 16, 19, 82, 88] [2, 9, 10, 19, 24, 39, 83, 91] [3, 7, 10, 12, 21, 26, 84, 93] [4, 8, 11, 13, 22, 27, 85, 94] [4, 5, 13, 15, 16, 32, 86, 97] [6, 18, 19, 32, 37, 52, 87, 100] [7, 16, 19, 21, 34, 39, 88, 102] [8, 17, 20, 22, 35, 40, 89, 103] [0, 3, 9, 18, 24, 57, 90, 105] [6, 10, 24, 26, 39, 44, 91, 107] [11, 25, 27, 40, 45, 60, 92, 108] [0, 1, 2, 12, 21, 26, 81, 93] [13, 22, 27, 29, 42, 47, 94, 110] [14, 23, 28, 30, 43, 48, 95, 111] [15, 27, 31, 32, 47, 50, 96, 113] [13, 16, 29, 32, 34, 52, 97, 115] [14, 17, 30, 33, 35, 53, 98, 116] [1, 7, 18, 31, 37, 66, 99, 118] [15, 19, 37, 39, 52, 57, 100, 120] [20, 38, 40, 53, 58, 69, 101, 121] [1, 5, 6, 21, 34, 39, 82, 102] [22, 35, 40, 42, 55, 60, 103, 123] [23, 36, 41, 43, 56, 61, 104, 124] [3, 12, 18, 24, 37, 44, 105, 125] [4, 13, 25, 38, 45, 72, 106, 126] [2, 6, 9, 19, 26, 44, 83, 107] [20, 27, 45, 47, 60, 63, 108, 128] [28, 46, 48, 61, 64, 74, 109, 129] [4, 8, 11, 29, 42, 47, 85, 110] [30, 43, 48, 49, 62, 65, 111, 130] [5, 16, 25, 31, 50, 63, 112, 131] [11, 32, 47, 50, 52, 66, 113, 133] [33, 48, 51, 53, 65, 67, 114, 134] [4, 5, 15, 29, 34, 52, 86, 115] [30, 35, 49, 53, 55, 69, 116, 136] [0, 3, 36, 54, 56, 70, 117, 137] [7, 21, 31, 37, 50, 57, 118, 138] [8, 22, 38, 51, 58, 76, 119, 139] [6, 15, 18, 32, 39, 57, 87, 120] [33, 40, 58, 60, 69, 72, 121, 141] [41, 59, 61, 70, 73, 78, 122, 142] [8, 17, 20, 42, 55, 60, 89, 123] [43, 56, 61, 62, 71, 74, 124, 143] [0, 9, 12, 37, 44, 57, 90, 125] [13, 29, 38, 45, 58, 63, 126, 144] [14, 30, 46, 59, 64, 79, 127, 145] [11, 20, 25, 40, 47, 63, 92, 128] [41, 48, 64, 65, 74, 75, 129, 146] [14, 23, 28, 49, 62, 65, 95, 130] [16, 25, 34, 45, 50, 66, 131, 147] [17, 35, 46, 51, 67, 75, 132, 148] [11, 15, 27, 31, 52, 66, 96, 133] [28, 53, 65, 67, 69, 76, 134, 150] [10, 26, 54, 68, 70, 77, 135, 151] [14, 17, 33, 49, 55, 69, 98, 136] [3, 12, 56, 70, 71, 78, 137, 152] [1, 18, 21, 50, 57, 66, 99, 138] [22, 42, 51, 58, 67, 72, 139, 153] [23, 43, 59, 68, 73, 80, 140, 154] [20, 33, 38, 53, 60, 72, 101, 141] [54, 61, 73, 74, 78, 79, 142, 155] [23, 36, 41, 62, 71, 74, 104, 143] [4, 25, 29, 58, 63, 72, 106, 144] [30, 49, 59, 64, 73, 75, 145, 156] [28, 41, 46, 61, 65, 75, 109, 146] [5, 31, 34, 45, 63, 66, 112, 147] [35, 46, 55, 64, 67, 76, 148, 157] [9, 36, 44, 56, 68, 77, 149, 158] [28, 33, 48, 51, 69, 76, 114, 150] [2, 26, 70, 77, 78, 80, 151, 159] [0, 12, 36, 54, 71, 78, 117, 152] [8, 38, 42, 67, 72, 76, 119, 153] [43, 62, 68, 73, 77, 79, 154, 160] [41, 54, 59, 70, 74, 79, 122, 155] [14, 46, 49, 73, 75, 79, 127, 156] [17, 51, 55, 64, 75, 76, 132, 157] [9, 24, 56, 71, 77, 80, 158, 161] [2, 10, 54, 68, 78, 80, 135, 159] [23, 59, 62, 77, 79, 80, 140, 160] [24, 36, 44, 68, 71, 80, 149, 161]
Code ID 162-54-3 · download JSON · raw on GitHub