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[[180,18,14]] d ≤
n
180
k
18
d
14
kd²/n
19.6
w
8

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Distance

d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[2, 9, 18, 29, 36, 37, 38, 47, 54, 63, 74, 81, 82, 83]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[96, 105, 115, 123, 124, 125, 133, 141, 150, 160, 168, 169, 170, 178]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction Coset-based two-block (2BGA) code. Group G = metacyclic C30 |x C6 (action r=11, order 180); non-normal subgroup H=C2; qubits indexed by cosets G/H. H_X=[L(a)|R(b)], H_Z=[R(b)^T|L(a)^T] with left/right coset actions; a,b element supports [0, 8, 34, 82] / [0, 54, 76, 162] found by simulated annealing. Weight-8 checks. A generalization of bivariate bicycle codes.
model Claude Claude Opus 4.8 (claimed, not verified)
builds on arXiv:2606.17268
date 2026-06-22
notes Distance is an UPPER BOUND (d<=14): witnesses are explicit decoder-found logical operators of weight 14 (syndrome-decoder method), cross-checked by randomized minimum-weight search (QDistRnd-style); both saturate at 14 over 6e5 trials, and the sub-threshold logical-error slope corroborates d~14. NOT certified exact. This is a coset 2BGA on a non-abelian metacyclic group -- a generalization of bivariate bicycle, not standard BB on a torus. Three more variants with identical [[180,16,14]] parameters exist. These codes are NOT 2D-local (no planar layout) and enter only the weight-8 any-connectivity track -- unlike the planar weight-8 codes of arXiv:2504.08887, which also occupy the 2d-local tracks. The comparison is on (n,k,d) at check weight <= 8 only. PROVISIONAL: d=14 here is an UNVERIFIED upper bound (the witness only proves d<=14); exact certification is pending. A prior Gurobi ILP + LP-decoding-cut attempt on this code stalled with the per-generator lower bound at 8, but the source paper certified comparable n<=168 weight-8 coset codes by integer programming, so certification may yet succeed. Independent residual-weight and QDistRnd searches find NO logical below weight 14. If the true distance is below 14 the kd^2/n figure drops accordingly; treat any top-of-board ranking as provisional until certified.
family 2BGA coset (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (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 90 · Z-checks 90
H_X (90 checks, sparse supports)
[0, 53, 77, 88, 90, 99, 146, 153] [1, 51, 75, 89, 91, 100, 144, 154] [2, 52, 76, 87, 92, 101, 145, 155] [1, 3, 56, 80, 93, 102, 149, 156] [2, 4, 54, 78, 94, 103, 147, 157] [0, 5, 55, 79, 95, 104, 148, 158] [4, 6, 59, 83, 96, 105, 152, 159] [5, 7, 57, 81, 97, 106, 150, 160] [3, 8, 58, 82, 98, 107, 151, 161] [7, 9, 62, 86, 99, 108, 155, 162] [8, 10, 60, 84, 100, 109, 153, 163] [6, 11, 61, 85, 101, 110, 154, 164] [10, 12, 65, 89, 102, 111, 158, 165] [11, 13, 63, 87, 103, 112, 156, 166] [9, 14, 64, 88, 104, 113, 157, 167] [2, 13, 15, 68, 105, 114, 161, 168] [0, 14, 16, 66, 106, 115, 159, 169] [1, 12, 17, 67, 107, 116, 160, 170] [5, 16, 18, 71, 108, 117, 164, 171] [3, 17, 19, 69, 109, 118, 162, 172] [4, 15, 20, 70, 110, 119, 163, 173] [8, 19, 21, 74, 111, 120, 167, 174] [6, 20, 22, 72, 112, 121, 165, 175] [7, 18, 23, 73, 113, 122, 166, 176] [11, 22, 24, 77, 114, 123, 170, 177] [9, 23, 25, 75, 115, 124, 168, 178] [10, 21, 26, 76, 116, 125, 169, 179] [14, 25, 27, 80, 90, 117, 126, 173] [12, 26, 28, 78, 91, 118, 127, 171] [13, 24, 29, 79, 92, 119, 128, 172] [17, 28, 30, 83, 93, 120, 129, 176] [15, 29, 31, 81, 94, 121, 130, 174] [16, 27, 32, 82, 95, 122, 131, 175] [20, 31, 33, 86, 96, 123, 132, 179] [18, 32, 34, 84, 97, 124, 133, 177] [19, 30, 35, 85, 98, 125, 134, 178] [23, 34, 36, 89, 92, 99, 126, 135] [21, 35, 37, 87, 90, 100, 127, 136] [22, 33, 38, 88, 91, 101, 128, 137] [2, 26, 37, 39, 95, 102, 129, 138] [0, 24, 38, 40, 93, 103, 130, 139] [1, 25, 36, 41, 94, 104, 131, 140] [5, 29, 40, 42, 98, 105, 132, 141] [3, 27, 41, 43, 96, 106, 133, 142] [4, 28, 39, 44, 97, 107, 134, 143] [8, 32, 43, 45, 101, 108, 135, 144] [6, 30, 44, 46, 99, 109, 136, 145] [7, 31, 42, 47, 100, 110, 137, 146] [11, 35, 46, 48, 104, 111, 138, 147] [9, 33, 47, 49, 102, 112, 139, 148] [10, 34, 45, 50, 103, 113, 140, 149] [14, 38, 49, 51, 107, 114, 141, 150] [12, 36, 50, 52, 105, 115, 142, 151] [13, 37, 48, 53, 106, 116, 143, 152] [17, 41, 52, 54, 110, 117, 144, 153] [15, 39, 53, 55, 108, 118, 145, 154] [16, 40, 51, 56, 109, 119, 146, 155] [20, 44, 55, 57, 113, 120, 147, 156] [18, 42, 56, 58, 111, 121, 148, 157] [19, 43, 54, 59, 112, 122, 149, 158] [23, 47, 58, 60, 116, 123, 150, 159] [21, 45, 59, 61, 114, 124, 151, 160] [22, 46, 57, 62, 115, 125, 152, 161] [26, 50, 61, 63, 119, 126, 153, 162] [24, 48, 62, 64, 117, 127, 154, 163] [25, 49, 60, 65, 118, 128, 155, 164] [29, 53, 64, 66, 122, 129, 156, 165] [27, 51, 65, 67, 120, 130, 157, 166] [28, 52, 63, 68, 121, 131, 158, 167] [32, 56, 67, 69, 125, 132, 159, 168] [30, 54, 68, 70, 123, 133, 160, 169] [31, 55, 66, 71, 124, 134, 161, 170] [35, 59, 70, 72, 128, 135, 162, 171] [33, 57, 71, 73, 126, 136, 163, 172] [34, 58, 69, 74, 127, 137, 164, 173] [38, 62, 73, 75, 131, 138, 165, 174] [36, 60, 74, 76, 129, 139, 166, 175] [37, 61, 72, 77, 130, 140, 167, 176] [41, 65, 76, 78, 134, 141, 168, 177] [39, 63, 77, 79, 132, 142, 169, 178] [40, 64, 75, 80, 133, 143, 170, 179] [44, 68, 79, 81, 90, 137, 144, 171] [42, 66, 80, 82, 91, 135, 145, 172] [43, 67, 78, 83, 92, 136, 146, 173] [47, 71, 82, 84, 93, 140, 147, 174] [45, 69, 83, 85, 94, 138, 148, 175] [46, 70, 81, 86, 95, 139, 149, 176] [50, 74, 85, 87, 96, 143, 150, 177] [48, 72, 86, 88, 97, 141, 151, 178] [49, 73, 84, 89, 98, 142, 152, 179]
H_Z (90 checks, sparse supports)
[0, 27, 37, 81, 90, 95, 106, 130] [1, 28, 38, 82, 91, 93, 107, 131] [2, 29, 36, 83, 92, 94, 105, 129] [3, 30, 40, 84, 93, 98, 109, 133] [4, 31, 41, 85, 94, 96, 110, 134] [5, 32, 39, 86, 95, 97, 108, 132] [6, 33, 43, 87, 96, 101, 112, 136] [7, 34, 44, 88, 97, 99, 113, 137] [8, 35, 42, 89, 98, 100, 111, 135] [0, 9, 36, 46, 99, 104, 115, 139] [1, 10, 37, 47, 100, 102, 116, 140] [2, 11, 38, 45, 101, 103, 114, 138] [3, 12, 39, 49, 102, 107, 118, 142] [4, 13, 40, 50, 103, 105, 119, 143] [5, 14, 41, 48, 104, 106, 117, 141] [6, 15, 42, 52, 105, 110, 121, 145] [7, 16, 43, 53, 106, 108, 122, 146] [8, 17, 44, 51, 107, 109, 120, 144] [9, 18, 45, 55, 108, 113, 124, 148] [10, 19, 46, 56, 109, 111, 125, 149] [11, 20, 47, 54, 110, 112, 123, 147] [12, 21, 48, 58, 111, 116, 127, 151] [13, 22, 49, 59, 112, 114, 128, 152] [14, 23, 50, 57, 113, 115, 126, 150] [15, 24, 51, 61, 114, 119, 130, 154] [16, 25, 52, 62, 115, 117, 131, 155] [17, 26, 53, 60, 116, 118, 129, 153] [18, 27, 54, 64, 117, 122, 133, 157] [19, 28, 55, 65, 118, 120, 134, 158] [20, 29, 56, 63, 119, 121, 132, 156] [21, 30, 57, 67, 120, 125, 136, 160] [22, 31, 58, 68, 121, 123, 137, 161] [23, 32, 59, 66, 122, 124, 135, 159] [24, 33, 60, 70, 123, 128, 139, 163] [25, 34, 61, 71, 124, 126, 140, 164] [26, 35, 62, 69, 125, 127, 138, 162] [27, 36, 63, 73, 126, 131, 142, 166] [28, 37, 64, 74, 127, 129, 143, 167] [29, 38, 65, 72, 128, 130, 141, 165] [30, 39, 66, 76, 129, 134, 145, 169] [31, 40, 67, 77, 130, 132, 146, 170] [32, 41, 68, 75, 131, 133, 144, 168] [33, 42, 69, 79, 132, 137, 148, 172] [34, 43, 70, 80, 133, 135, 149, 173] [35, 44, 71, 78, 134, 136, 147, 171] [36, 45, 72, 82, 135, 140, 151, 175] [37, 46, 73, 83, 136, 138, 152, 176] [38, 47, 74, 81, 137, 139, 150, 174] [39, 48, 75, 85, 138, 143, 154, 178] [40, 49, 76, 86, 139, 141, 155, 179] [41, 50, 77, 84, 140, 142, 153, 177] [42, 51, 78, 88, 91, 141, 146, 157] [43, 52, 79, 89, 92, 142, 144, 158] [44, 53, 80, 87, 90, 143, 145, 156] [1, 45, 54, 81, 94, 144, 149, 160] [2, 46, 55, 82, 95, 145, 147, 161] [0, 47, 56, 83, 93, 146, 148, 159] [4, 48, 57, 84, 97, 147, 152, 163] [5, 49, 58, 85, 98, 148, 150, 164] [3, 50, 59, 86, 96, 149, 151, 162] [7, 51, 60, 87, 100, 150, 155, 166] [8, 52, 61, 88, 101, 151, 153, 167] [6, 53, 62, 89, 99, 152, 154, 165] [0, 10, 54, 63, 103, 153, 158, 169] [1, 11, 55, 64, 104, 154, 156, 170] [2, 9, 56, 65, 102, 155, 157, 168] [3, 13, 57, 66, 106, 156, 161, 172] [4, 14, 58, 67, 107, 157, 159, 173] [5, 12, 59, 68, 105, 158, 160, 171] [6, 16, 60, 69, 109, 159, 164, 175] [7, 17, 61, 70, 110, 160, 162, 176] [8, 15, 62, 71, 108, 161, 163, 174] [9, 19, 63, 72, 112, 162, 167, 178] [10, 20, 64, 73, 113, 163, 165, 179] [11, 18, 65, 74, 111, 164, 166, 177] [12, 22, 66, 75, 91, 115, 165, 170] [13, 23, 67, 76, 92, 116, 166, 168] [14, 21, 68, 77, 90, 114, 167, 169] [15, 25, 69, 78, 94, 118, 168, 173] [16, 26, 70, 79, 95, 119, 169, 171] [17, 24, 71, 80, 93, 117, 170, 172] [18, 28, 72, 81, 97, 121, 171, 176] [19, 29, 73, 82, 98, 122, 172, 174] [20, 27, 74, 83, 96, 120, 173, 175] [21, 31, 75, 84, 100, 124, 174, 179] [22, 32, 76, 85, 101, 125, 175, 177] [23, 30, 77, 86, 99, 123, 176, 178] [24, 34, 78, 87, 92, 103, 127, 177] [25, 35, 79, 88, 90, 104, 128, 178] [26, 33, 80, 89, 91, 102, 126, 179]
raw submission JSON