[[160,6,9]] · QEC Challenge

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[[160,6,9]] d =

160

kd²/n

3.038

X/Z

1.11

0.0105

4.1231

layers

swaps

258

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Distance

X/Z asymmetry 1.11 · d_X = 9, d_Z = 10 · 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 9 · witness weight 9 (claimed exact)

witness operator (support, 9 qubits)

[2, 56, 68, 77, 93, 96, 104, 115, 158]

d_Z 10 · witness weight 10 (claimed exact)

witness operator (support, 10 qubits)

[32, 34, 36, 38, 51, 112, 114, 116, 118, 121]

certificate exact, d = 9 · scipy/HiGHS cutoff IP

X: no logical < 9 exists; Z: no logical < 10 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 2–6 (mean 4.682) · H_Z 2–6 (mean 4.889)

qubit degrees H_X 1–5 (mean 2.575) · H_Z 1–5 (mean 2.475)

trapping sets H_X (1,1)×27 (2,0)×12 (3,1)×43 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 27 (1,2): 38 (1,3): 73 (1,4): 20 (1,5): 2 (2,0): 12 (2,1): 40 (2,2): 84 (2,3): 159 (2,4): 375 (2,5): 142 (2,6): 35 (2,7): 5 (3,1): 43 (3,2): 267 (3,3): 677 (3,4): 1284 (3,5): 2620 (3,6): 1285 (3,7): 748 (3,8): 243 (3,9): 63 (3,10): 12 (3,11): 1

trapping sets H_Z (1,1)×38 (2,0)×18 (3,1)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 38 (1,2): 35 (1,3): 67 (1,4): 13 (1,5): 7 (2,0): 18 (2,1): 64 (2,2): 91 (2,3): 177 (2,4): 309 (2,5): 103 (2,6): 48 (2,7): 16 (2,8): 5 (3,1): 56 (3,2): 407 (3,3): 745 (3,4): 1479 (3,5): 2115 (3,6): 975 (3,7): 662 (3,8): 307 (3,9): 114 (3,10): 32 (3,11): 5

witness diameter X 9.2195 · Z 7.2801 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

X checkZ checkqubit site (80)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

routing cost 258 nearest-neighbor SWAPs per round in total, at most 3 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @FarLab and @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction f = 1 + xy + x^-1 y², g = 1 + x^-1 y + x^-2 y^-1; annulus-type boundary engine, (8,10) lattice.

model Claude Opus 4.8 (claimed, not verified)

builds on arXiv:2504.08887

date 2026-06-16

family bivariate bicycle (a tag, not a ranking)

locality 2D-local bilayer (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 88 (max weight 6) · Z-checks 81 (max weight 6)

H_X (88 checks, sparse supports)

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

H_Z (81 checks, sparse supports)

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

Code ID 160-6-9 · download JSON · raw on GitHub