[[192,64,3]] · QEC Challenge

[[192,64,3]] d =

192

kd²/n

3.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 6, w_Z = 6 (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)

[26, 98, 103]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[55, 80, 93]

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 6 · H_Z 6

qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×96 (2,2)×288 (3,0)×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 (1,4): 96 (2,2): 288 (2,4): 576 (2,6): 192 (3,0): 160 (3,2): 960 (3,4): 2400 (3,6): 2208 (3,8): 384

trapping sets H_Z (1,2)×96 (2,2)×288 (3,0)×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 (1,4): 96 (2,2): 288 (2,4): 576 (2,6): 192 (3,0): 160 (3,2): 960 (3,4): 2400 (3,6): 2208 (3,8): 384

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(96,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order96_k64.txt, line 1. nonidentity supports a=[7], b=[2, 7, 12].

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

[[192,64,3]] — two-block group-algebra code on SmallGroup(96,1)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 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_wt6wtL2_order96_k64.txt, line 1. This row is SmallGroup(96,1) with nonidentity GAP supports a=[7], b=[2, 7, 12]. 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=64), 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 (d=3) 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(96,1), 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=[7], b=[2, 7, 12] 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) = 64. 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 6) · Z-checks 96 (max weight 6)

H_X (96 checks, sparse supports)

[0, 22, 96, 118, 169, 185] [1, 11, 96, 97, 102, 129] [2, 40, 97, 98, 107, 136] [3, 44, 99, 103, 122, 140] [4, 46, 100, 119, 142, 146] [5, 47, 101, 143, 144, 170] [0, 6, 96, 102, 169, 191] [7, 26, 98, 103, 111, 150] [8, 29, 99, 104, 114, 154] [9, 31, 100, 105, 116, 156] [10, 32, 101, 106, 117, 157] [11, 33, 96, 97, 107, 118] [12, 65, 104, 108, 125, 161] [13, 67, 105, 109, 127, 163] [14, 68, 106, 110, 128, 164] [2, 15, 97, 98, 111, 129] [16, 70, 112, 120, 148, 166] [17, 71, 113, 121, 149, 167] [3, 18, 99, 103, 114, 150] [19, 72, 115, 145, 168, 171] [4, 20, 100, 116, 119, 172] [5, 21, 101, 117, 144, 186] [6, 22, 102, 118, 185, 191] [23, 50, 108, 119, 132, 172] [24, 52, 109, 120, 134, 174] [25, 53, 110, 121, 135, 175] [26, 54, 98, 103, 122, 136] [27, 56, 112, 123, 138, 177] [28, 57, 113, 124, 139, 178] [29, 58, 99, 104, 125, 140] [30, 59, 115, 126, 141, 179] [31, 60, 100, 105, 127, 142] [32, 61, 101, 106, 128, 143] [1, 33, 102, 107, 118, 129] [34, 85, 123, 130, 152, 181] [35, 86, 124, 131, 153, 182] [12, 36, 104, 108, 132, 154] [37, 87, 126, 133, 155, 183] [13, 38, 105, 109, 134, 156] [14, 39, 106, 110, 135, 157] [15, 40, 107, 111, 129, 136] [41, 88, 137, 147, 173, 184] [16, 42, 112, 120, 138, 174] [17, 43, 113, 121, 139, 175] [18, 44, 114, 122, 140, 150] [19, 45, 115, 141, 145, 187] [20, 46, 116, 142, 146, 172] [21, 47, 117, 143, 170, 186] [48, 74, 130, 144, 159, 186] [49, 75, 131, 145, 160, 187] [50, 76, 108, 119, 146, 161] [51, 77, 133, 147, 162, 188] [52, 78, 109, 120, 148, 163] [53, 79, 110, 121, 149, 164] [7, 54, 111, 122, 136, 150] [55, 80, 137, 151, 165, 189] [56, 81, 112, 123, 152, 166] [57, 82, 113, 124, 153, 167] [8, 58, 114, 125, 140, 154] [59, 83, 115, 126, 155, 168] [9, 60, 116, 127, 142, 156] [10, 61, 117, 128, 143, 157] [62, 94, 151, 158, 176, 190] [34, 63, 123, 130, 159, 177] [35, 64, 124, 131, 160, 178] [36, 65, 125, 132, 154, 161] [37, 66, 126, 133, 162, 179] [38, 67, 127, 134, 156, 163] [39, 68, 128, 135, 157, 164] [41, 69, 137, 147, 165, 188] [42, 70, 138, 148, 166, 174] [43, 71, 139, 149, 167, 175] [45, 72, 141, 168, 171, 187] [73, 89, 158, 169, 180, 191] [74, 90, 130, 144, 170, 181] [75, 91, 131, 145, 171, 182] [23, 76, 132, 146, 161, 172] [77, 92, 133, 147, 173, 183] [24, 78, 134, 148, 163, 174] [25, 79, 135, 149, 164, 175] [80, 93, 137, 151, 176, 184] [27, 81, 138, 152, 166, 177] [28, 82, 139, 153, 167, 178] [30, 83, 141, 155, 168, 179] [62, 84, 151, 158, 180, 189] [63, 85, 152, 159, 177, 181] [64, 86, 153, 160, 178, 182] [66, 87, 155, 162, 179, 183] [69, 88, 165, 173, 184, 188] [89, 95, 158, 169, 185, 190] [48, 90, 159, 170, 181, 186] [49, 91, 160, 171, 182, 187] [51, 92, 162, 173, 183, 188] [55, 93, 165, 176, 184, 189] [84, 94, 176, 180, 189, 190] [73, 95, 180, 185, 190, 191]

H_Z (96 checks, sparse supports)

[0, 1, 6, 11, 96, 102] [1, 2, 11, 15, 97, 129] [2, 7, 15, 26, 98, 111] [3, 8, 18, 29, 99, 114] [4, 9, 20, 31, 100, 116] [5, 10, 21, 32, 101, 117] [1, 6, 22, 33, 102, 118] [3, 7, 18, 26, 103, 150] [8, 12, 29, 36, 104, 154] [9, 13, 31, 38, 105, 156] [10, 14, 32, 39, 106, 157] [2, 11, 33, 40, 97, 107] [12, 23, 36, 50, 108, 132] [13, 24, 38, 52, 109, 134] [14, 25, 39, 53, 110, 135] [7, 15, 40, 54, 111, 136] [16, 27, 42, 56, 112, 138] [17, 28, 43, 57, 113, 139] [8, 18, 44, 58, 114, 140] [19, 30, 45, 59, 115, 141] [9, 20, 46, 60, 116, 142] [10, 21, 47, 61, 117, 143] [0, 11, 22, 33, 96, 118] [4, 20, 23, 50, 119, 172] [16, 24, 42, 52, 120, 174] [17, 25, 43, 53, 121, 175] [3, 26, 44, 54, 103, 122] [27, 34, 56, 63, 123, 177] [28, 35, 57, 64, 124, 178] [12, 29, 58, 65, 104, 125] [30, 37, 59, 66, 126, 179] [13, 31, 60, 67, 105, 127] [14, 32, 61, 68, 106, 128] [1, 15, 33, 40, 107, 129] [34, 48, 63, 74, 130, 159] [35, 49, 64, 75, 131, 160] [23, 36, 65, 76, 132, 161] [37, 51, 66, 77, 133, 162] [24, 38, 67, 78, 134, 163] [25, 39, 68, 79, 135, 164] [2, 26, 40, 54, 98, 136] [41, 55, 69, 80, 137, 165] [27, 42, 70, 81, 138, 166] [28, 43, 71, 82, 139, 167] [3, 29, 44, 58, 99, 140] [30, 45, 72, 83, 141, 168] [4, 31, 46, 60, 100, 142] [5, 32, 47, 61, 101, 143] [5, 21, 48, 74, 144, 186] [19, 45, 49, 75, 145, 187] [4, 46, 50, 76, 119, 146] [41, 51, 69, 77, 147, 188] [16, 52, 70, 78, 120, 148] [17, 53, 71, 79, 121, 149] [7, 18, 44, 54, 122, 150] [55, 62, 80, 84, 151, 189] [34, 56, 81, 85, 123, 152] [35, 57, 82, 86, 124, 153] [8, 36, 58, 65, 125, 154] [37, 59, 83, 87, 126, 155] [9, 38, 60, 67, 127, 156] [10, 39, 61, 68, 128, 157] [62, 73, 84, 89, 158, 180] [48, 63, 85, 90, 159, 181] [49, 64, 86, 91, 160, 182] [12, 50, 65, 76, 108, 161] [51, 66, 87, 92, 162, 183] [13, 52, 67, 78, 109, 163] [14, 53, 68, 79, 110, 164] [55, 69, 88, 93, 165, 184] [16, 56, 70, 81, 112, 166] [17, 57, 71, 82, 113, 167] [19, 59, 72, 83, 115, 168] [0, 6, 73, 89, 169, 191] [5, 47, 74, 90, 144, 170] [19, 72, 75, 91, 145, 171] [20, 23, 46, 76, 146, 172] [41, 77, 88, 92, 147, 173] [24, 42, 70, 78, 148, 174] [25, 43, 71, 79, 149, 175] [62, 80, 93, 94, 151, 176] [27, 63, 81, 85, 152, 177] [28, 64, 82, 86, 153, 178] [30, 66, 83, 87, 155, 179] [73, 84, 94, 95, 180, 190] [34, 74, 85, 90, 130, 181] [35, 75, 86, 91, 131, 182] [37, 77, 87, 92, 133, 183] [41, 80, 88, 93, 137, 184] [0, 22, 89, 95, 169, 185] [21, 47, 48, 90, 170, 186] [45, 49, 72, 91, 171, 187] [51, 69, 88, 92, 173, 188] [55, 84, 93, 94, 176, 189] [62, 89, 94, 95, 158, 190] [6, 22, 73, 95, 185, 191]

Code ID 192-64-3 · download JSON · raw on GitHub