[[148,4,18]] · QEC Challenge

[[148,4,18]] d ≤

148

kd²/n

8.757

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[5, 19, 23, 38, 40, 43, 54, 60, 65, 68, 77, 91, 101, 103, 104, 108, 109, 113]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[8, 25, 27, 54, 56, 57, 65, 71, 77, 84, 90, 92, 104, 112, 118, 122, 123, 145]

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)×148 (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): 148 (2,6): 1776 (3,4): 296 (3,6): 4070 (3,8): 30562 (3,10): 2664

trapping sets H_Z (1,4)×148 (2,4)×148 (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): 148 (2,6): 1776 (3,4): 296 (3,6): 4070 (3,8): 30562 (3,10): 2664

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_k4.txt, line 12. nonidentity supports a=[2, 3, 10], b=[7, 21, 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

[[148,4,18]] — 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 = 8.76.

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_k4.txt, line 12. This row is SmallGroup(74,1) with nonidentity GAP supports a=[2, 3, 10], b=[7, 21, 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 18, 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, 10], b=[7, 21, 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) = 4. 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, 9, 72, 74, 96, 128, 142] [0, 1, 3, 66, 75, 97, 129, 143] [0, 2, 3, 11, 76, 98, 130, 144] [2, 3, 5, 68, 77, 99, 131, 145] [2, 4, 5, 13, 78, 100, 132, 146] [4, 5, 7, 70, 79, 101, 133, 147] [4, 6, 7, 15, 74, 80, 102, 134] [6, 7, 9, 72, 75, 81, 103, 135] [6, 8, 9, 17, 76, 82, 104, 136] [0, 8, 9, 11, 77, 83, 105, 137] [8, 10, 11, 19, 78, 84, 106, 138] [2, 10, 11, 13, 79, 85, 107, 139] [10, 12, 13, 21, 80, 86, 108, 140] [4, 12, 13, 15, 81, 87, 109, 141] [12, 14, 15, 23, 82, 88, 110, 142] [6, 14, 15, 17, 83, 89, 111, 143] [14, 16, 17, 25, 84, 90, 112, 144] [8, 16, 17, 19, 85, 91, 113, 145] [16, 18, 19, 27, 86, 92, 114, 146] [10, 18, 19, 21, 87, 93, 115, 147] [18, 20, 21, 29, 74, 88, 94, 116] [12, 20, 21, 23, 75, 89, 95, 117] [20, 22, 23, 31, 76, 90, 96, 118] [14, 22, 23, 25, 77, 91, 97, 119] [22, 24, 25, 33, 78, 92, 98, 120] [16, 24, 25, 27, 79, 93, 99, 121] [24, 26, 27, 35, 80, 94, 100, 122] [18, 26, 27, 29, 81, 95, 101, 123] [26, 28, 29, 37, 82, 96, 102, 124] [20, 28, 29, 31, 83, 97, 103, 125] [28, 30, 31, 39, 84, 98, 104, 126] [22, 30, 31, 33, 85, 99, 105, 127] [30, 32, 33, 41, 86, 100, 106, 128] [24, 32, 33, 35, 87, 101, 107, 129] [32, 34, 35, 43, 88, 102, 108, 130] [26, 34, 35, 37, 89, 103, 109, 131] [34, 36, 37, 45, 90, 104, 110, 132] [28, 36, 37, 39, 91, 105, 111, 133] [36, 38, 39, 47, 92, 106, 112, 134] [30, 38, 39, 41, 93, 107, 113, 135] [38, 40, 41, 49, 94, 108, 114, 136] [32, 40, 41, 43, 95, 109, 115, 137] [40, 42, 43, 51, 96, 110, 116, 138] [34, 42, 43, 45, 97, 111, 117, 139] [42, 44, 45, 53, 98, 112, 118, 140] [36, 44, 45, 47, 99, 113, 119, 141] [44, 46, 47, 55, 100, 114, 120, 142] [38, 46, 47, 49, 101, 115, 121, 143] [46, 48, 49, 57, 102, 116, 122, 144] [40, 48, 49, 51, 103, 117, 123, 145] [48, 50, 51, 59, 104, 118, 124, 146] [42, 50, 51, 53, 105, 119, 125, 147] [50, 52, 53, 61, 74, 106, 120, 126] [44, 52, 53, 55, 75, 107, 121, 127] [52, 54, 55, 63, 76, 108, 122, 128] [46, 54, 55, 57, 77, 109, 123, 129] [54, 56, 57, 65, 78, 110, 124, 130] [48, 56, 57, 59, 79, 111, 125, 131] [56, 58, 59, 67, 80, 112, 126, 132] [50, 58, 59, 61, 81, 113, 127, 133] [58, 60, 61, 69, 82, 114, 128, 134] [52, 60, 61, 63, 83, 115, 129, 135] [60, 62, 63, 71, 84, 116, 130, 136] [54, 62, 63, 65, 85, 117, 131, 137] [62, 64, 65, 73, 86, 118, 132, 138] [56, 64, 65, 67, 87, 119, 133, 139] [1, 64, 66, 67, 88, 120, 134, 140] [58, 66, 67, 69, 89, 121, 135, 141] [3, 66, 68, 69, 90, 122, 136, 142] [60, 68, 69, 71, 91, 123, 137, 143] [5, 68, 70, 71, 92, 124, 138, 144] [62, 70, 71, 73, 93, 125, 139, 145] [7, 70, 72, 73, 94, 126, 140, 146] [1, 64, 72, 73, 95, 127, 141, 147]

H_Z (74 checks, sparse supports)

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

Code ID 148-4-18 · download JSON · raw on GitHub