[[162,44,2]] · QEC Challenge

[[162,44,2]] d =

162

kd²/n

1.086

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 7, w_Z = 7 (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)

[98, 150]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[20, 58]

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 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×81 (2,0)×81 (3,0)×27 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,3): 81 (1,4): 81 (2,0): 81 (2,4): 162 (2,5): 972 (2,6): 162 (3,0): 27 (3,3): 27 (3,4): 1215 (3,5): 1944 (3,6): 5022 (3,7): 7776 (3,8): 324 (3,9): 486

trapping sets H_Z (1,3)×81 (2,0)×81 (3,0)×27 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,3): 81 (1,4): 81 (2,0): 81 (2,4): 162 (2,5): 972 (2,6): 162 (3,0): 27 (3,3): 27 (3,4): 1215 (3,5): 1944 (3,6): 5022 (3,7): 7776 (3,8): 324 (3,9): 486

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_wt7wtL3_order81_k44.txt, line 1. nonidentity supports a=[3, 10], b=[2, 4, 8].

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

[[162,44,2]] — two-block group-algebra code on SmallGroup(81,3)

Direction & hypothesis

Target: unrestricted weight-8 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 = 1.09.

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_wt7wtL3_order81_k44.txt, line 1. This row is SmallGroup(81,3) with nonidentity GAP supports a=[3, 10], b=[2, 4, 8]. 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=44), 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(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=[3, 10], b=[2, 4, 8] 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) = 44. 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 7) · Z-checks 81 (max weight 7)

H_X (81 checks, sparse supports)

[0, 2, 9, 81, 93, 117, 152] [1, 19, 57, 81, 82, 93, 102] [0, 2, 9, 83, 107, 151, 159] [3, 10, 24, 81, 84, 117, 137] [4, 11, 25, 85, 86, 110, 115] [5, 50, 52, 82, 86, 102, 115] [6, 21, 37, 87, 91, 107, 120] [7, 18, 39, 81, 82, 84, 88] [8, 40, 72, 85, 89, 110, 123] [0, 2, 9, 90, 125, 149, 158] [3, 10, 24, 83, 91, 135, 159] [4, 11, 25, 92, 113, 128, 133] [12, 26, 44, 84, 93, 137, 152] [13, 27, 45, 85, 86, 94, 97] [14, 28, 46, 95, 98, 130, 136] [15, 16, 66, 96, 100, 120, 133] [15, 16, 66, 82, 86, 88, 97] [17, 67, 69, 89, 98, 123, 136] [7, 18, 39, 90, 99, 105, 138] [1, 19, 57, 83, 87, 100, 107] [20, 42, 58, 101, 108, 128, 141] [6, 21, 37, 84, 88, 93, 102] [22, 38, 60, 85, 89, 94, 103] [23, 61, 79, 95, 104, 130, 143] [3, 10, 24, 90, 105, 158, 161] [4, 11, 25, 106, 112, 131, 144] [12, 26, 44, 91, 107, 135, 151] [13, 27, 45, 92, 96, 108, 133] [14, 28, 46, 109, 134, 146, 150] [29, 47, 63, 94, 97, 110, 115] [30, 48, 64, 95, 98, 111, 116] [31, 32, 34, 99, 112, 118, 147] [31, 32, 34, 87, 96, 113, 120] [33, 35, 76, 114, 121, 141, 150] [31, 32, 34, 88, 97, 102, 115] [33, 35, 76, 89, 98, 103, 116] [36, 77, 78, 104, 117, 143, 152] [6, 21, 37, 99, 105, 118, 125] [22, 38, 60, 106, 119, 126, 153] [7, 18, 39, 83, 91, 100, 120] [8, 40, 72, 92, 101, 121, 128] [41, 62, 73, 122, 129, 146, 155] [20, 42, 58, 94, 103, 110, 123] [43, 59, 74, 95, 104, 111, 124] [12, 26, 44, 105, 125, 149, 161] [13, 27, 45, 106, 126, 131, 147] [14, 28, 46, 127, 132, 148, 156] [29, 47, 63, 96, 108, 113, 128] [30, 48, 64, 109, 114, 129, 150] [49, 65, 75, 111, 116, 130, 136] [5, 50, 52, 112, 118, 131, 138] [51, 53, 55, 119, 132, 139, 157] [5, 50, 52, 87, 100, 113, 133] [51, 53, 55, 101, 114, 134, 141] [54, 56, 80, 135, 142, 155, 159] [51, 53, 55, 103, 116, 123, 136] [54, 56, 80, 104, 117, 124, 137] [1, 19, 57, 90, 118, 125, 138] [20, 42, 58, 119, 126, 139, 144] [43, 59, 74, 127, 140, 145, 160] [22, 38, 60, 92, 108, 121, 141] [23, 61, 79, 109, 122, 142, 146] [41, 62, 73, 111, 124, 130, 143] [29, 47, 63, 112, 126, 144, 147] [30, 48, 64, 127, 145, 148, 157] [49, 65, 75, 114, 129, 134, 146] [15, 16, 66, 99, 131, 138, 147] [17, 67, 69, 132, 139, 148, 153] [68, 70, 71, 140, 149, 154, 161] [17, 67, 69, 101, 121, 134, 150] [68, 70, 71, 122, 135, 151, 155] [68, 70, 71, 124, 137, 143, 152] [8, 40, 72, 106, 139, 144, 153] [41, 62, 73, 140, 145, 154, 156] [43, 59, 74, 109, 129, 142, 155] [49, 65, 75, 132, 145, 156, 157] [33, 35, 76, 119, 148, 153, 157] [36, 77, 78, 149, 154, 158, 160] [36, 77, 78, 122, 142, 151, 159] [23, 61, 79, 127, 154, 156, 160] [54, 56, 80, 140, 158, 160, 161]

H_Z (81 checks, sparse supports)

[0, 1, 3, 7, 81, 83, 90] [1, 5, 7, 16, 82, 100, 138] [2, 10, 19, 39, 81, 83, 90] [3, 7, 12, 21, 84, 91, 105] [4, 8, 13, 22, 85, 92, 106] [4, 5, 13, 16, 86, 131, 133] [6, 19, 32, 52, 87, 102, 118] [7, 16, 21, 34, 88, 99, 120] [8, 17, 22, 35, 89, 121, 153] [9, 18, 24, 57, 81, 83, 90] [6, 10, 26, 39, 84, 91, 105] [11, 27, 40, 60, 85, 92, 106] [0, 1, 12, 21, 93, 107, 125] [13, 22, 29, 42, 94, 108, 126] [14, 23, 30, 43, 95, 109, 127] [15, 27, 32, 47, 96, 97, 147] [13, 16, 29, 34, 96, 97, 147] [14, 17, 30, 35, 98, 148, 150] [18, 31, 37, 66, 88, 99, 120] [15, 19, 39, 52, 82, 100, 138] [20, 40, 53, 69, 101, 123, 139] [1, 5, 21, 34, 87, 102, 118] [22, 35, 42, 55, 103, 119, 141] [23, 36, 43, 56, 104, 142, 160] [18, 24, 37, 44, 84, 91, 105] [25, 38, 45, 72, 85, 92, 106] [2, 6, 19, 26, 93, 107, 125] [20, 27, 47, 60, 94, 108, 126] [28, 48, 61, 74, 95, 109, 127] [4, 8, 29, 42, 110, 128, 144] [30, 43, 49, 62, 111, 129, 145] [25, 31, 50, 63, 112, 113, 115] [11, 32, 47, 52, 112, 113, 115] [33, 48, 53, 65, 114, 116, 157] [4, 5, 29, 34, 112, 113, 115] [30, 35, 49, 55, 114, 116, 157] [0, 3, 36, 56, 117, 158, 159] [31, 37, 50, 57, 87, 102, 118] [38, 51, 58, 76, 103, 119, 141] [6, 15, 32, 39, 88, 99, 120] [33, 40, 60, 69, 89, 121, 153] [41, 61, 70, 78, 122, 143, 154] [8, 17, 42, 55, 101, 123, 139] [43, 56, 62, 71, 124, 140, 155] [9, 37, 44, 57, 93, 107, 125] [38, 45, 58, 63, 94, 108, 126] [46, 59, 64, 79, 95, 109, 127] [11, 20, 40, 47, 110, 128, 144] [41, 48, 65, 74, 111, 129, 145] [14, 23, 49, 62, 130, 146, 156] [25, 45, 50, 66, 86, 131, 133] [46, 51, 67, 75, 132, 134, 136] [11, 15, 27, 52, 86, 131, 133] [28, 53, 65, 69, 132, 134, 136] [10, 26, 54, 70, 135, 137, 161] [14, 17, 49, 55, 132, 134, 136] [3, 12, 56, 71, 135, 137, 161] [18, 50, 57, 66, 82, 100, 138] [51, 58, 67, 72, 101, 123, 139] [59, 68, 73, 80, 124, 140, 155] [20, 33, 53, 60, 103, 119, 141] [54, 61, 74, 78, 104, 142, 160] [23, 36, 62, 71, 122, 143, 154] [25, 58, 63, 72, 110, 128, 144] [59, 64, 73, 75, 111, 129, 145] [28, 41, 61, 65, 130, 146, 156] [31, 45, 63, 66, 96, 97, 147] [46, 64, 67, 76, 98, 148, 150] [9, 44, 68, 77, 149, 151, 152] [28, 33, 48, 69, 98, 148, 150] [2, 26, 70, 78, 149, 151, 152] [0, 12, 36, 71, 149, 151, 152] [38, 67, 72, 76, 89, 121, 153] [68, 73, 77, 79, 122, 143, 154] [41, 54, 70, 74, 124, 140, 155] [46, 73, 75, 79, 130, 146, 156] [51, 64, 75, 76, 114, 116, 157] [9, 24, 77, 80, 117, 158, 159] [2, 10, 54, 78, 117, 158, 159] [59, 77, 79, 80, 104, 142, 160] [24, 44, 68, 80, 135, 137, 161]

Code ID 162-44-2 · download JSON · raw on GitHub