[[144,72,2]] · QEC Challenge

[[144,72,2]] d =

144

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)

[73, 79]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[96, 114]

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)×72 (2,0)×72 (3,2)×1080 (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 (1,6): 72 (2,0): 72 (2,4): 432 (2,8): 432 (3,2): 1080 (3,6): 4392 (3,10): 3456 (3,14): 288

trapping sets H_Z (1,2)×72 (2,0)×72 (3,2)×1080 (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 (1,6): 72 (2,0): 72 (2,4): 432 (2,8): 432 (3,2): 1080 (3,6): 4392 (3,10): 3456 (3,14): 288

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(72,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order72_k72.txt, line 1. nonidentity supports a=[4], b=[2, 4, 5, 8, 14].

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

[[144,72,2]] — two-block group-algebra code on SmallGroup(72,1)

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 = 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_order72_k72.txt, line 1. This row is SmallGroup(72,1) with nonidentity GAP supports a=[4], b=[2, 4, 5, 8, 14]. 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=72), 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(72,1), 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=[4], b=[2, 4, 5, 8, 14] 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) = 72. 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 8) · Z-checks 72 (max weight 8)

H_X (72 checks, sparse supports)

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

H_Z (72 checks, sparse supports)

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

Code ID 144-72-2 · download JSON · raw on GitHub