[[156,78,2]] · QEC Challenge

[[156,78,2]] d =

156

kd²/n

2.0

X/Z

Share this result

Distance

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

witness operator (support, 2 qubits)

[92, 95]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[115, 118]

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 8 · H_Z 8

qubit degrees H_X 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)

trapping sets H_X (1,2)×78 (2,0)×78 (3,2)×1170 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 78 (1,6): 78 (2,0): 78 (2,4): 468 (2,8): 468 (3,2): 1170 (3,6): 4654 (3,10): 4056 (3,14): 312

trapping sets H_Z (1,2)×78 (2,0)×78 (3,2)×1170 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 78 (1,6): 78 (2,0): 78 (2,4): 468 (2,8): 468 (3,2): 1170 (3,6): 4654 (3,10): 4056 (3,14): 312

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(78,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order78_k78.txt, line 1. nonidentity supports a=[2], b=[2, 3, 4, 5, 6].

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

[[156,78,2]] — two-block group-algebra code on SmallGroup(78,2)

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 = 2.00.

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_wt8wtL2_order78_k78.txt, line 1. This row is SmallGroup(78,2) with nonidentity GAP supports a=[2], b=[2, 3, 4, 5, 6]. 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=78), 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 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(78,2), 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], b=[2, 3, 4, 5, 6] 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 = 156 - rank(HX) - rank(HZ) = 78. 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 78 (max weight 8) · Z-checks 78 (max weight 8)

H_X (78 checks, sparse supports)

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

H_Z (78 checks, sparse supports)

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

Code ID 156-78-2 · download JSON · raw on GitHub