[[160,48,2]] · QEC Challenge

[[160,48,2]] d =

160

kd²/n

1.2

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)

[122, 130]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[129, 137]

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)×80 (2,0)×80 (3,0)×64 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 80 (1,4): 80 (2,0): 80 (2,2): 64 (2,4): 512 (2,6): 320 (3,0): 64 (3,2): 288 (3,4): 2400 (3,6): 4224 (3,8): 2304

trapping sets H_Z (1,2)×80 (2,0)×40 (3,2)×512 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 80 (1,4): 80 (2,0): 40 (2,2): 64 (2,4): 752 (3,2): 512 (3,4): 3696 (3,6): 1824 (3,8): 160

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(80,4), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order80_k48.txt, line 1. nonidentity supports a=[2], b=[2, 13, 20].

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

[[160,48,2]] — two-block group-algebra code on SmallGroup(80,4)

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 = 1.20.

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_order80_k48.txt, line 1. This row is SmallGroup(80,4) with nonidentity GAP supports a=[2], b=[2, 13, 20]. 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=48), 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(80,4), 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, 13, 20] 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 = 160 - rank(HX) - rank(HZ) = 48. 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 80 (max weight 6) · Z-checks 80 (max weight 6)

H_X (80 checks, sparse supports)

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

H_Z (80 checks, sparse supports)

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

Code ID 160-48-2 · download JSON · raw on GitHub