[[156,56,2]] · QEC Challenge

[[156,56,2]] d =

156

kd²/n

1.436

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)

[125, 135]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[61, 66]

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 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×78 (2,0)×117 (3,3)×494 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 78 (1,4): 78 (2,0): 117 (2,4): 156 (2,5): 936 (3,3): 494 (3,4): 1404 (3,5): 1872 (3,6): 4212 (3,7): 4680

trapping sets H_Z (1,3)×78 (2,0)×117 (3,3)×494 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 78 (1,4): 78 (2,0): 117 (2,4): 156 (2,5): 936 (3,3): 494 (3,4): 1404 (3,5): 1872 (3,6): 4212 (3,7): 4680

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(78,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt7wtL3_order78_k56.txt, line 1. nonidentity supports a=[3, 7], b=[2, 4, 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

[[156,56,2]] — two-block group-algebra code on SmallGroup(78,1)

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 = 1.44.

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_wt7wtL3_order78_k56.txt, line 1. This row is SmallGroup(78,1) with nonidentity GAP supports a=[3, 7], b=[2, 4, 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=56), 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(78,1), 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=[3, 7], b=[2, 4, 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) = 56. 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, 2, 6, 78, 79, 83, 146] [1, 4, 9, 78, 79, 81, 149] [0, 2, 6, 80, 82, 88, 151] [3, 7, 12, 78, 79, 81, 149] [1, 4, 9, 80, 82, 85, 153] [5, 10, 15, 78, 79, 83, 146] [0, 2, 6, 84, 87, 93, 154] [3, 7, 12, 80, 82, 85, 153] [8, 13, 18, 81, 86, 143, 149] [1, 4, 9, 84, 87, 90, 155] [5, 10, 15, 80, 82, 88, 151] [11, 16, 21, 83, 89, 140, 146] [3, 7, 12, 84, 87, 90, 155] [8, 13, 18, 85, 91, 148, 153] [14, 19, 24, 86, 92, 137, 143] [5, 10, 15, 84, 87, 93, 154] [11, 16, 21, 88, 94, 145, 151] [17, 22, 27, 89, 95, 134, 140] [8, 13, 18, 90, 96, 152, 155] [14, 19, 24, 91, 97, 142, 148] [20, 25, 30, 92, 98, 131, 137] [11, 16, 21, 93, 99, 150, 154] [17, 22, 27, 94, 100, 139, 145] [23, 28, 33, 95, 101, 128, 134] [14, 19, 24, 96, 102, 147, 152] [20, 25, 30, 97, 103, 136, 142] [26, 31, 36, 98, 104, 125, 131] [17, 22, 27, 99, 105, 144, 150] [23, 28, 33, 100, 106, 133, 139] [29, 34, 39, 101, 107, 122, 128] [20, 25, 30, 102, 108, 141, 147] [26, 31, 36, 103, 109, 130, 136] [32, 37, 42, 104, 110, 119, 125] [23, 28, 33, 105, 111, 138, 144] [29, 34, 39, 106, 112, 127, 133] [35, 40, 45, 107, 113, 116, 122] [26, 31, 36, 108, 114, 135, 141] [32, 37, 42, 109, 115, 124, 130] [38, 43, 48, 110, 113, 116, 119] [29, 34, 39, 111, 117, 132, 138] [35, 40, 45, 112, 118, 121, 127] [41, 46, 51, 110, 113, 116, 119] [32, 37, 42, 114, 120, 129, 135] [38, 43, 48, 115, 118, 121, 124] [44, 49, 54, 107, 113, 116, 122] [35, 40, 45, 117, 123, 126, 132] [41, 46, 51, 115, 118, 121, 124] [47, 52, 57, 104, 110, 119, 125] [38, 43, 48, 120, 123, 126, 129] [44, 49, 54, 112, 118, 121, 127] [50, 55, 60, 101, 107, 122, 128] [41, 46, 51, 120, 123, 126, 129] [47, 52, 57, 109, 115, 124, 130] [53, 58, 63, 98, 104, 125, 131] [44, 49, 54, 117, 123, 126, 132] [50, 55, 60, 106, 112, 127, 133] [56, 61, 66, 95, 101, 128, 134] [47, 52, 57, 114, 120, 129, 135] [53, 58, 63, 103, 109, 130, 136] [59, 64, 69, 92, 98, 131, 137] [50, 55, 60, 111, 117, 132, 138] [56, 61, 66, 100, 106, 133, 139] [62, 67, 72, 89, 95, 134, 140] [53, 58, 63, 108, 114, 135, 141] [59, 64, 69, 97, 103, 136, 142] [65, 70, 74, 86, 92, 137, 143] [56, 61, 66, 105, 111, 138, 144] [62, 67, 72, 94, 100, 139, 145] [68, 73, 76, 83, 89, 140, 146] [59, 64, 69, 102, 108, 141, 147] [65, 70, 74, 91, 97, 142, 148] [71, 75, 77, 81, 86, 143, 149] [62, 67, 72, 99, 105, 144, 150] [68, 73, 76, 88, 94, 145, 151] [65, 70, 74, 96, 102, 147, 152] [71, 75, 77, 85, 91, 148, 153] [68, 73, 76, 93, 99, 150, 154] [71, 75, 77, 90, 96, 152, 155]

H_Z (78 checks, sparse supports)

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

Code ID 156-56-2 · download JSON · raw on GitHub