[[148,2,19]] · QEC Challenge

[[148,2,19]] d ≤

148

kd²/n

4.878

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · 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 19 · witness weight 19 (claimed upper_bound)

witness operator (support, 19 qubits)

[20, 24, 27, 38, 40, 48, 73, 90, 94, 96, 97, 99, 101, 104, 118, 122, 137, 138, 140]

d_Z 19 · witness weight 19 (claimed upper_bound)

witness operator (support, 19 qubits)

[15, 16, 17, 19, 39, 43, 52, 54, 56, 57, 81, 83, 85, 105, 116, 117, 127, 134, 135]

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 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 4 · H_Z 4

trapping sets H_X (1,4)×148 (2,4)×185 (3,4)×296 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 148 (2,4): 185 (2,6): 1702 (3,4): 296 (3,6): 4736 (3,8): 28638 (3,10): 2368

trapping sets H_Z (1,4)×148 (2,4)×185 (3,4)×296 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 148 (2,4): 185 (2,6): 1702 (3,4): 296 (3,6): 4736 (3,8): 28638 (3,10): 2368

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(74,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order74_k2.txt, line 14. nonidentity supports a=[2, 3, 5], b=[11, 29, 53].

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

[[148,2,19]] — two-block group-algebra code on SmallGroup(74,1)

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 = 4.88.

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_wt8wtL4_order74_k2.txt, line 14. This row is SmallGroup(74,1) with nonidentity GAP supports a=[2, 3, 5], b=[11, 29, 53]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 19, 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(74,1), 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=[2, 3, 5], b=[11, 29, 53] 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 = 148 - rank(HX) - rank(HZ) = 2. 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 74 (max weight 8) · Z-checks 74 (max weight 8)

H_X (74 checks, sparse supports)

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

H_Z (74 checks, sparse supports)

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

Code ID 148-2-19 · download JSON · raw on GitHub