[[164,4,19]] · QEC Challenge

[[164,4,19]] d ≤

164

kd²/n

8.805

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)

[3, 8, 14, 22, 42, 47, 53, 63, 75, 84, 103, 105, 139, 140, 147, 148, 150, 154, 156]

d_Z 19 · witness weight 19 (claimed upper_bound)

witness operator (support, 19 qubits)

[6, 8, 9, 12, 17, 19, 82, 84, 86, 96, 100, 102, 114, 118, 120, 126, 132, 142, 150]

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)×164 (2,4)×246 (3,4)×164 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 164 (2,4): 246 (2,6): 1804 (3,4): 164 (3,6): 6806 (3,8): 28618 (3,10): 2296

trapping sets H_Z (1,4)×164 (2,4)×246 (3,4)×164 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 164 (2,4): 246 (2,6): 1804 (3,4): 164 (3,6): 6806 (3,8): 28618 (3,10): 2296

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(82,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order82_k4.txt, line 13. nonidentity supports a=[2, 3, 8], b=[7, 23, 55].

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

[[164,4,19]] — two-block group-algebra code on SmallGroup(82,1)

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 = 8.80.

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_order82_k4.txt, line 13. This row is SmallGroup(82,1) with nonidentity GAP supports a=[2, 3, 8], b=[7, 23, 55]. 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 ≤ 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(82,1), 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, 8], b=[7, 23, 55] 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 = 164 - 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 82 (max weight 8) · Z-checks 82 (max weight 8)

H_X (82 checks, sparse supports)

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

H_Z (82 checks, sparse supports)

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

Code ID 164-4-19 · download JSON · raw on GitHub