[[156,52,2]] · QEC Challenge

[[156,52,2]] d =

156

kd²/n

1.333

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 7, w_Z = 7 (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)

[13, 16]

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 7 · H_Z 7

qubit degrees H_X 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)

trapping sets H_X (1,2)×78 (2,0)×39 (3,2)×312 (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 (1,5): 78 (2,0): 39 (2,3): 156 (2,4): 78 (2,5): 468 (2,6): 39 (2,8): 468 (3,2): 312 (3,3): 936 (3,4): 234 (3,5): 1404 (3,6): 1404 (3,7): 1170 (3,8): 6162 (3,9): 702 (3,11): 3822

trapping sets H_Z (1,2)×78 (2,0)×39 (3,2)×312 (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 (1,5): 78 (2,0): 39 (2,3): 156 (2,4): 78 (2,5): 468 (2,6): 39 (2,8): 468 (3,2): 312 (3,3): 936 (3,4): 234 (3,5): 1404 (3,6): 1404 (3,7): 1170 (3,8): 6162 (3,9): 702 (3,11): 3822

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_wt7wtL2_order78_k52.txt, line 1. nonidentity supports a=[2], b=[2, 3, 4, 67].

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

[[156,52,2]] — two-block group-algebra code on SmallGroup(78,2)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.33.

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_wt7wtL2_order78_k52.txt, line 1. This row is SmallGroup(78,2) with nonidentity GAP supports a=[2], b=[2, 3, 4, 67]. 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=52), 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(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, 67] 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) = 52. 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 7) · Z-checks 78 (max weight 7)

H_X (78 checks, sparse supports)

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

H_Z (78 checks, sparse supports)

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

Code ID 156-52-2 · download JSON · raw on GitHub