[[192,72,2]] · QEC Challenge

[[192,72,2]] d =

192

kd²/n

1.5

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[142, 163]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[6, 15]

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 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 (2,0)×48 (3,2)×1536 (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 (2,0): 48 (2,2): 192 (2,4): 672 (3,2): 1536 (3,4): 2304 (3,6): 1440 (3,8): 192

trapping sets H_Z (1,2)×96 (2,0)×96 (3,0)×192 (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 (2,0): 96 (2,2): 192 (2,4): 384 (2,6): 384 (3,0): 192 (3,2): 288 (3,4): 2592 (3,6): 3456 (3,8): 2304

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(96,48), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order96_k72.txt, line 1. nonidentity supports a=[3], b=[3, 9, 24].

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

[[192,72,2]] — two-block group-algebra code on SmallGroup(96,48)

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 = 1.50.

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_k72.txt, line 1. This row is SmallGroup(96,48) with nonidentity GAP supports a=[3], b=[3, 9, 24]. 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=72), 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(96,48), 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], b=[3, 9, 24] 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) = 72. 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, 2, 96, 98, 178, 191] [1, 24, 97, 103, 112, 158] [0, 2, 96, 98, 178, 191] [3, 12, 99, 108, 129, 175] [4, 13, 100, 109, 187, 189] [5, 14, 101, 110, 123, 169] [6, 15, 102, 111, 152, 185] [7, 9, 97, 103, 112, 158] [8, 49, 96, 104, 109, 119] [7, 9, 105, 120, 130, 137] [10, 52, 106, 121, 138, 180] [11, 53, 107, 122, 139, 181] [3, 12, 99, 108, 129, 175] [4, 13, 100, 109, 187, 189] [5, 14, 101, 110, 123, 169] [6, 15, 102, 111, 152, 185] [16, 34, 112, 130, 156, 188] [17, 35, 113, 131, 150, 157] [18, 36, 97, 114, 120, 132] [19, 37, 106, 115, 133, 148] [20, 38, 116, 134, 144, 151] [21, 39, 117, 135, 170, 176] [22, 40, 118, 136, 153, 186] [23, 28, 96, 104, 109, 119] [1, 24, 105, 120, 130, 137] [25, 31, 106, 121, 138, 180] [26, 32, 107, 122, 139, 181] [27, 73, 99, 123, 131, 144] [23, 28, 98, 100, 124, 145] [29, 76, 101, 125, 134, 146] [30, 77, 102, 126, 135, 147] [25, 31, 127, 148, 159, 165] [26, 32, 128, 149, 160, 166] [33, 79, 129, 150, 167, 190] [16, 34, 112, 130, 156, 188] [17, 35, 113, 131, 150, 157] [18, 36, 97, 114, 120, 132] [19, 37, 106, 115, 133, 148] [20, 38, 116, 134, 144, 151] [21, 39, 117, 135, 170, 176] [22, 40, 118, 136, 153, 186] [41, 62, 137, 158, 174, 179] [42, 63, 104, 138, 145, 159] [43, 64, 125, 139, 160, 172] [44, 65, 103, 105, 140, 161] [45, 66, 121, 127, 141, 162] [46, 67, 107, 142, 149, 163] [47, 68, 143, 164, 171, 177] [48, 55, 99, 123, 131, 144] [8, 49, 98, 100, 124, 145] [50, 58, 101, 125, 134, 146] [51, 59, 102, 126, 135, 147] [10, 52, 127, 148, 159, 165] [11, 53, 128, 149, 160, 166] [54, 61, 129, 150, 167, 190] [48, 55, 108, 113, 151, 169] [56, 89, 114, 152, 161, 170] [57, 90, 115, 153, 162, 171] [50, 58, 110, 116, 154, 172] [51, 59, 111, 117, 155, 173] [60, 92, 118, 156, 164, 174] [54, 61, 157, 175, 182, 184] [41, 62, 137, 158, 174, 179] [42, 63, 104, 138, 145, 159] [43, 64, 125, 139, 160, 172] [44, 65, 103, 105, 140, 161] [45, 66, 121, 127, 141, 162] [46, 67, 107, 142, 149, 163] [47, 68, 143, 164, 171, 177] [69, 84, 119, 124, 165, 180] [70, 85, 146, 154, 166, 181] [71, 86, 126, 167, 173, 182] [72, 87, 122, 128, 168, 183] [27, 73, 108, 113, 151, 169] [74, 80, 114, 152, 161, 170] [75, 81, 115, 153, 162, 171] [29, 76, 110, 116, 154, 172] [30, 77, 111, 117, 155, 173] [78, 83, 118, 156, 164, 174] [33, 79, 157, 175, 182, 184] [74, 80, 132, 140, 176, 185] [75, 81, 133, 141, 177, 186] [82, 95, 142, 178, 183, 187] [78, 83, 136, 143, 179, 188] [69, 84, 119, 124, 165, 180] [70, 85, 146, 154, 166, 181] [71, 86, 126, 167, 173, 182] [72, 87, 122, 128, 168, 183] [88, 94, 147, 155, 184, 190] [56, 89, 132, 140, 176, 185] [57, 90, 133, 141, 177, 186] [91, 93, 142, 178, 183, 187] [60, 92, 136, 143, 179, 188] [91, 93, 163, 168, 189, 191] [88, 94, 147, 155, 184, 190] [82, 95, 163, 168, 189, 191]

H_Z (96 checks, sparse supports)

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

Code ID 192-72-2 · download JSON · raw on GitHub