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[[192,14,4]] d =
n
192
k
14
d
4
kd²/n
1.167
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[103, 107, 119, 125]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[6, 18, 40, 65]
certificate exact, d = 4 · scipy/HiGHS cutoff IP
X: no logical < 4 exists; Z: no logical < 4 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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×96 (2,2)×96 (3,2)×160 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 96 (1,3): 96 (2,2): 96 (2,3): 576 (2,4): 288 (3,2): 160 (3,3): 1728 (3,4): 2976 (3,5): 1440 (3,6): 576 (3,7): 96
trapping sets H_Z (1,2)×96 (2,2)×96 (3,2)×160 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 96 (1,3): 96 (2,2): 96 (2,3): 576 (2,4): 288 (3,2): 160 (3,3): 1728 (3,4): 2976 (3,5): 1440 (3,6): 576 (3,7): 96

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(96,65), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt5wtL2_order96_k14.txt, line 2. nonidentity supports a=[15], b=[11, 14].
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

[[192,14,4]] — two-block group-algebra code on SmallGroup(96,65)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-5 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.17.

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_wt5wtL2_order96_k14.txt, line 2. This row is SmallGroup(96,65) with nonidentity GAP supports a=[15], b=[11, 14]. 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=14), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 4, not an exact certificate. The published distance (d=4) 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(96,65), 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=[15], b=[11, 14] 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 = 192 - rank(HX) - rank(HZ) = 14. 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 96 (max weight 5) · Z-checks 96 (max weight 5)
H_X (96 checks, sparse supports)
[0, 35, 96, 146, 158] [1, 50, 97, 102, 169] [2, 5, 98, 107, 115] [3, 14, 99, 122, 133] [4, 64, 100, 108, 186] [5, 12, 101, 119, 190] [6, 65, 102, 172, 180] [7, 11, 103, 111, 126] [8, 26, 104, 114, 147] [9, 91, 105, 119, 142] [10, 76, 96, 106, 191] [11, 23, 107, 118, 185] [12, 17, 108, 125, 137] [13, 21, 96, 109, 156] [0, 14, 97, 110, 168] [15, 22, 111, 129, 141] [16, 39, 98, 112, 174] [2, 17, 103, 113, 183] [18, 40, 114, 150, 162] [19, 94, 115, 130, 171] [20, 86, 116, 161, 169] [21, 34, 117, 131, 191] [22, 36, 118, 145, 181] [23, 29, 119, 132, 151] [24, 61, 97, 120, 163] [25, 33, 98, 121, 179] [1, 26, 122, 136, 155] [27, 79, 103, 123, 166] [28, 54, 99, 124, 188] [7, 29, 125, 140, 173] [30, 89, 117, 126, 144] [31, 75, 115, 127, 172] [32, 74, 128, 145, 168] [33, 49, 101, 129, 186] [34, 43, 99, 130, 177] [3, 35, 104, 131, 184] [36, 44, 132, 154, 165] [37, 72, 100, 128, 133] [38, 47, 118, 126, 134] [4, 39, 101, 135, 179] [6, 40, 106, 136, 142] [41, 87, 109, 137, 149] [42, 68, 136, 138, 147] [13, 43, 110, 139, 188] [15, 44, 121, 140, 163] [45, 85, 141, 144, 182] [46, 84, 142, 159, 187] [47, 63, 132, 143, 185] [48, 82, 104, 144, 181] [49, 58, 108, 145, 189] [8, 50, 146, 161, 176] [51, 59, 105, 135, 147] [32, 52, 129, 133, 148] [31, 53, 106, 149, 183] [10, 54, 110, 150, 156] [55, 77, 120, 139, 151] [53, 56, 137, 150, 152] [52, 57, 121, 153, 184] [25, 58, 113, 154, 174] [59, 73, 100, 155, 187] [60, 95, 143, 156, 170] [48, 61, 141, 146, 157] [62, 88, 112, 153, 158] [63, 71, 140, 151, 159] [16, 64, 113, 160, 189] [18, 65, 124, 161, 166] [46, 66, 105, 143, 162] [45, 67, 116, 157, 163] [20, 68, 102, 155, 164] [67, 69, 120, 164, 165] [66, 70, 134, 166, 175] [38, 71, 111, 167, 173] [62, 72, 160, 168, 170] [73, 80, 123, 160, 169] [57, 74, 154, 158, 170] [56, 75, 124, 171, 190] [28, 76, 131, 172, 177] [30, 77, 109, 157, 173] [78, 83, 127, 164, 174] [9, 79, 107, 162, 175] [51, 80, 112, 175, 176] [81, 92, 148, 167, 177] [24, 82, 122, 165, 178] [83, 90, 116, 171, 179] [70, 84, 123, 167, 180] [69, 85, 138, 178, 181] [42, 86, 114, 176, 182] [19, 87, 117, 127, 183] [37, 88, 135, 148, 184] [55, 89, 130, 178, 185] [90, 93, 152, 182, 186] [27, 91, 125, 180, 187] [60, 92, 128, 134, 188] [78, 93, 138, 149, 189] [41, 94, 139, 152, 190] [81, 95, 153, 159, 191]
H_Z (96 checks, sparse supports)
[0, 10, 13, 96, 110] [1, 14, 24, 97, 122] [2, 16, 25, 98, 113] [3, 28, 34, 99, 131] [4, 37, 59, 100, 135] [5, 33, 39, 98, 101] [1, 6, 68, 102, 136] [7, 17, 27, 103, 125] [8, 35, 48, 104, 146] [9, 51, 66, 105, 175] [10, 40, 53, 106, 150] [2, 11, 79, 103, 107] [4, 12, 49, 101, 108] [13, 41, 77, 109, 139] [14, 43, 54, 99, 110] [7, 15, 71, 111, 140] [16, 62, 80, 112, 160] [17, 58, 64, 108, 113] [8, 18, 86, 114, 161] [2, 19, 31, 115, 183] [20, 67, 83, 116, 164] [21, 30, 87, 109, 117] [11, 22, 38, 111, 118] [5, 9, 23, 107, 119] [24, 55, 69, 120, 178] [25, 44, 57, 121, 154] [3, 26, 82, 104, 122] [27, 73, 84, 123, 187] [28, 65, 75, 124, 172] [12, 29, 91, 119, 125] [7, 30, 38, 126, 173] [31, 78, 87, 127, 149] [32, 37, 92, 128, 148] [15, 33, 52, 121, 129] [19, 34, 89, 117, 130] [21, 35, 76, 96, 131] [23, 36, 47, 118, 132] [3, 37, 52, 133, 184] [38, 70, 92, 134, 167] [39, 51, 88, 112, 135] [26, 40, 42, 114, 136] [12, 41, 56, 137, 190] [42, 85, 93, 138, 182] [43, 55, 94, 130, 139] [29, 44, 63, 132, 140] [15, 45, 61, 141, 163] [9, 40, 46, 142, 162] [47, 60, 66, 134, 143] [30, 45, 48, 144, 157] [22, 32, 49, 129, 145] [0, 50, 61, 97, 146] [8, 42, 51, 147, 176] [52, 81, 88, 148, 153] [41, 53, 93, 149, 152] [18, 54, 56, 124, 150] [23, 55, 63, 151, 185] [56, 90, 94, 152, 171] [57, 62, 95, 153, 170] [36, 58, 74, 145, 154] [26, 59, 68, 147, 155] [13, 54, 60, 156, 188] [61, 67, 77, 120, 157] [0, 62, 74, 158, 168] [46, 63, 95, 143, 159] [64, 72, 73, 100, 160] [20, 50, 65, 102, 161] [18, 66, 79, 162, 166] [24, 44, 67, 163, 165] [68, 69, 78, 138, 164] [36, 69, 82, 165, 181] [27, 65, 70, 166, 180] [71, 81, 84, 159, 167] [14, 32, 72, 133, 168] [1, 20, 73, 155, 169] [60, 72, 74, 128, 170] [19, 75, 83, 127, 171] [6, 31, 76, 106, 172] [29, 71, 77, 151, 173] [16, 58, 78, 174, 189] [70, 79, 80, 123, 175] [50, 80, 86, 169, 176] [34, 76, 81, 177, 191] [82, 85, 89, 144, 178] [25, 39, 83, 174, 179] [6, 84, 91, 142, 180] [22, 48, 85, 141, 181] [45, 86, 90, 116, 182] [17, 53, 87, 137, 183] [35, 57, 88, 158, 184] [11, 47, 89, 126, 185] [4, 33, 90, 179, 186] [46, 59, 91, 105, 187] [28, 43, 92, 177, 188] [49, 64, 93, 186, 189] [5, 75, 94, 115, 190] [10, 21, 95, 156, 191]
Code ID 192-14-4 · download JSON · raw on GitHub