[[176,88,2]] · QEC Challenge

[[176,88,2]] d =

176

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)

[161, 168]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[41, 48]

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)×88 (2,0)×88 (3,2)×1320 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 88 (1,6): 88 (2,0): 88 (2,4): 528 (2,8): 528 (3,2): 1320 (3,6): 5368 (3,10): 4224 (3,14): 352

trapping sets H_Z (1,2)×88 (2,0)×88 (3,2)×1320 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 88 (1,6): 88 (2,0): 88 (2,4): 528 (2,8): 528 (3,2): 1320 (3,6): 5368 (3,10): 4224 (3,14): 352

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(88,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order88_k88.txt, line 1. nonidentity supports a=[4], b=[2, 4, 5, 7, 11].

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

[[176,88,2]] — two-block group-algebra code on SmallGroup(88,1)

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_order88_k88.txt, line 1. This row is SmallGroup(88,1) with nonidentity GAP supports a=[4], b=[2, 4, 5, 7, 11]. 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=88), 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(88,1), 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, 7, 11] 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 = 176 - rank(HX) - rank(HZ) = 88. 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 88 (max weight 8) · Z-checks 88 (max weight 8)

H_X (88 checks, sparse supports)

[0, 3, 88, 91, 93, 100, 163, 170] [1, 6, 88, 89, 91, 94, 167, 173] [2, 8, 89, 90, 94, 96, 169, 174] [0, 3, 88, 91, 93, 100, 163, 170] [4, 10, 88, 91, 92, 98, 172, 175] [5, 12, 90, 93, 96, 100, 172, 175] [1, 6, 88, 89, 91, 94, 167, 173] [7, 14, 89, 94, 95, 102, 163, 170] [2, 8, 89, 90, 94, 96, 169, 174] [9, 16, 90, 96, 97, 104, 167, 173] [4, 10, 88, 91, 92, 98, 172, 175] [11, 18, 92, 98, 99, 106, 165, 171] [5, 12, 90, 93, 96, 100, 172, 175] [13, 20, 93, 100, 101, 108, 169, 174] [7, 14, 89, 94, 95, 102, 163, 170] [15, 22, 95, 102, 103, 110, 155, 162] [9, 16, 90, 96, 97, 104, 167, 173] [17, 24, 97, 104, 105, 112, 159, 166] [11, 18, 92, 98, 99, 106, 165, 171] [19, 26, 99, 106, 107, 114, 157, 164] [13, 20, 93, 100, 101, 108, 169, 174] [21, 28, 101, 108, 109, 116, 161, 168] [15, 22, 95, 102, 103, 110, 155, 162] [23, 30, 103, 110, 111, 118, 147, 154] [17, 24, 97, 104, 105, 112, 159, 166] [25, 32, 105, 112, 113, 120, 151, 158] [19, 26, 99, 106, 107, 114, 157, 164] [27, 34, 107, 114, 115, 122, 149, 156] [21, 28, 101, 108, 109, 116, 161, 168] [29, 36, 109, 116, 117, 124, 153, 160] [23, 30, 103, 110, 111, 118, 147, 154] [31, 38, 111, 118, 119, 126, 139, 146] [25, 32, 105, 112, 113, 120, 151, 158] [33, 40, 113, 120, 121, 128, 143, 150] [27, 34, 107, 114, 115, 122, 149, 156] [35, 42, 115, 122, 123, 130, 141, 148] [29, 36, 109, 116, 117, 124, 153, 160] [37, 44, 117, 124, 125, 132, 145, 152] [31, 38, 111, 118, 119, 126, 139, 146] [39, 46, 119, 126, 127, 131, 134, 138] [33, 40, 113, 120, 121, 128, 143, 150] [41, 48, 121, 128, 129, 135, 136, 142] [35, 42, 115, 122, 123, 130, 141, 148] [43, 50, 123, 130, 131, 133, 138, 140] [37, 44, 117, 124, 125, 132, 145, 152] [45, 52, 125, 132, 133, 137, 140, 144] [39, 46, 119, 126, 127, 131, 134, 138] [47, 54, 123, 127, 130, 134, 135, 142] [41, 48, 121, 128, 129, 135, 136, 142] [49, 56, 127, 129, 134, 136, 137, 144] [43, 50, 123, 130, 131, 133, 138, 140] [51, 58, 125, 131, 132, 138, 139, 146] [45, 52, 125, 132, 133, 137, 140, 144] [53, 60, 129, 133, 136, 140, 141, 148] [47, 54, 123, 127, 130, 134, 135, 142] [55, 62, 115, 122, 135, 142, 143, 150] [49, 56, 127, 129, 134, 136, 137, 144] [57, 64, 119, 126, 137, 144, 145, 152] [51, 58, 125, 131, 132, 138, 139, 146] [59, 66, 117, 124, 139, 146, 147, 154] [53, 60, 129, 133, 136, 140, 141, 148] [61, 68, 121, 128, 141, 148, 149, 156] [55, 62, 115, 122, 135, 142, 143, 150] [63, 70, 107, 114, 143, 150, 151, 158] [57, 64, 119, 126, 137, 144, 145, 152] [65, 72, 111, 118, 145, 152, 153, 160] [59, 66, 117, 124, 139, 146, 147, 154] [67, 74, 109, 116, 147, 154, 155, 162] [61, 68, 121, 128, 141, 148, 149, 156] [69, 76, 113, 120, 149, 156, 157, 164] [63, 70, 107, 114, 143, 150, 151, 158] [71, 78, 99, 106, 151, 158, 159, 166] [65, 72, 111, 118, 145, 152, 153, 160] [73, 80, 103, 110, 153, 160, 161, 168] [67, 74, 109, 116, 147, 154, 155, 162] [75, 82, 101, 108, 155, 162, 163, 170] [69, 76, 113, 120, 149, 156, 157, 164] [77, 83, 105, 112, 157, 164, 165, 171] [71, 78, 99, 106, 151, 158, 159, 166] [79, 85, 92, 98, 159, 166, 167, 173] [73, 80, 103, 110, 153, 160, 161, 168] [81, 86, 95, 102, 161, 168, 169, 174] [75, 82, 101, 108, 155, 162, 163, 170] [77, 83, 105, 112, 157, 164, 165, 171] [84, 87, 97, 104, 165, 171, 172, 175] [79, 85, 92, 98, 159, 166, 167, 173] [81, 86, 95, 102, 161, 168, 169, 174] [84, 87, 97, 104, 165, 171, 172, 175]

H_Z (88 checks, sparse supports)

[0, 1, 3, 4, 6, 10, 88, 91] [1, 2, 6, 7, 8, 14, 89, 94] [2, 5, 8, 9, 12, 16, 90, 96] [0, 1, 3, 4, 6, 10, 88, 91] [4, 10, 11, 18, 79, 85, 92, 98] [0, 3, 5, 12, 13, 20, 93, 100] [1, 2, 6, 7, 8, 14, 89, 94] [7, 14, 15, 22, 81, 86, 95, 102] [2, 5, 8, 9, 12, 16, 90, 96] [9, 16, 17, 24, 84, 87, 97, 104] [4, 10, 11, 18, 79, 85, 92, 98] [11, 18, 19, 26, 71, 78, 99, 106] [0, 3, 5, 12, 13, 20, 93, 100] [13, 20, 21, 28, 75, 82, 101, 108] [7, 14, 15, 22, 81, 86, 95, 102] [15, 22, 23, 30, 73, 80, 103, 110] [9, 16, 17, 24, 84, 87, 97, 104] [17, 24, 25, 32, 77, 83, 105, 112] [11, 18, 19, 26, 71, 78, 99, 106] [19, 26, 27, 34, 63, 70, 107, 114] [13, 20, 21, 28, 75, 82, 101, 108] [21, 28, 29, 36, 67, 74, 109, 116] [15, 22, 23, 30, 73, 80, 103, 110] [23, 30, 31, 38, 65, 72, 111, 118] [17, 24, 25, 32, 77, 83, 105, 112] [25, 32, 33, 40, 69, 76, 113, 120] [19, 26, 27, 34, 63, 70, 107, 114] [27, 34, 35, 42, 55, 62, 115, 122] [21, 28, 29, 36, 67, 74, 109, 116] [29, 36, 37, 44, 59, 66, 117, 124] [23, 30, 31, 38, 65, 72, 111, 118] [31, 38, 39, 46, 57, 64, 119, 126] [25, 32, 33, 40, 69, 76, 113, 120] [33, 40, 41, 48, 61, 68, 121, 128] [27, 34, 35, 42, 55, 62, 115, 122] [35, 42, 43, 47, 50, 54, 123, 130] [29, 36, 37, 44, 59, 66, 117, 124] [37, 44, 45, 51, 52, 58, 125, 132] [31, 38, 39, 46, 57, 64, 119, 126] [39, 46, 47, 49, 54, 56, 127, 134] [33, 40, 41, 48, 61, 68, 121, 128] [41, 48, 49, 53, 56, 60, 129, 136] [35, 42, 43, 47, 50, 54, 123, 130] [39, 43, 46, 50, 51, 58, 131, 138] [37, 44, 45, 51, 52, 58, 125, 132] [43, 45, 50, 52, 53, 60, 133, 140] [39, 46, 47, 49, 54, 56, 127, 134] [41, 47, 48, 54, 55, 62, 135, 142] [41, 48, 49, 53, 56, 60, 129, 136] [45, 49, 52, 56, 57, 64, 137, 144] [39, 43, 46, 50, 51, 58, 131, 138] [31, 38, 51, 58, 59, 66, 139, 146] [43, 45, 50, 52, 53, 60, 133, 140] [35, 42, 53, 60, 61, 68, 141, 148] [41, 47, 48, 54, 55, 62, 135, 142] [33, 40, 55, 62, 63, 70, 143, 150] [45, 49, 52, 56, 57, 64, 137, 144] [37, 44, 57, 64, 65, 72, 145, 152] [31, 38, 51, 58, 59, 66, 139, 146] [23, 30, 59, 66, 67, 74, 147, 154] [35, 42, 53, 60, 61, 68, 141, 148] [27, 34, 61, 68, 69, 76, 149, 156] [33, 40, 55, 62, 63, 70, 143, 150] [25, 32, 63, 70, 71, 78, 151, 158] [37, 44, 57, 64, 65, 72, 145, 152] [29, 36, 65, 72, 73, 80, 153, 160] [23, 30, 59, 66, 67, 74, 147, 154] [15, 22, 67, 74, 75, 82, 155, 162] [27, 34, 61, 68, 69, 76, 149, 156] [19, 26, 69, 76, 77, 83, 157, 164] [25, 32, 63, 70, 71, 78, 151, 158] [17, 24, 71, 78, 79, 85, 159, 166] [29, 36, 65, 72, 73, 80, 153, 160] [21, 28, 73, 80, 81, 86, 161, 168] [15, 22, 67, 74, 75, 82, 155, 162] [0, 3, 7, 14, 75, 82, 163, 170] [19, 26, 69, 76, 77, 83, 157, 164] [11, 18, 77, 83, 84, 87, 165, 171] [17, 24, 71, 78, 79, 85, 159, 166] [1, 6, 9, 16, 79, 85, 167, 173] [21, 28, 73, 80, 81, 86, 161, 168] [2, 8, 13, 20, 81, 86, 169, 174] [0, 3, 7, 14, 75, 82, 163, 170] [11, 18, 77, 83, 84, 87, 165, 171] [4, 5, 10, 12, 84, 87, 172, 175] [1, 6, 9, 16, 79, 85, 167, 173] [2, 8, 13, 20, 81, 86, 169, 174] [4, 5, 10, 12, 84, 87, 172, 175]

Code ID 176-88-2 · download JSON · raw on GitHub