[[144,42,2]] · QEC Challenge

[[144,42,2]] d =

144

kd²/n

1.167

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)

[85, 93]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[49, 57]

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)×72 (2,0)×72 (3,0)×48 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 72 (1,4): 72 (2,0): 72 (2,2): 48 (2,4): 480 (2,6): 288 (3,0): 48 (3,2): 264 (3,4): 2184 (3,6): 3936 (3,8): 2112

trapping sets H_Z (1,2)×72 (2,0)×36 (3,2)×384 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 72 (1,4): 72 (2,0): 36 (2,2): 48 (2,4): 696 (3,2): 384 (3,4): 3504 (3,6): 1704 (3,8): 144

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(72,28), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order72_k42.txt, line 1. nonidentity supports a=[2], b=[2, 29, 39].

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

[[144,42,2]] — two-block group-algebra code on SmallGroup(72,28)

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 = 1.17.

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_order72_k42.txt, line 1. This row is SmallGroup(72,28) with nonidentity GAP supports a=[2], b=[2, 29, 39]. 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=42), 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(72,28), 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, 29, 39] 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 = 144 - rank(HX) - rank(HZ) = 42. 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 72 (max weight 6) · Z-checks 72 (max weight 6)

H_X (72 checks, sparse supports)

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

H_Z (72 checks, sparse supports)

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

Code ID 144-42-2 · download JSON · raw on GitHub