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[[180,20,14]] d ≤
n
180
k
20
d
14
kd²/n
21.778
w
8

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Distance

d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[5, 12, 23, 39, 40, 50, 59, 67, 68, 93, 124, 147, 155, 166]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[2, 13, 18, 44, 72, 92, 101, 108, 110, 126, 137, 154, 155, 173]
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, 50, 146, 172] / [0, 18, 55, 147] 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, 8, 19, 67, 90, 108, 155, 171] [1, 6, 20, 68, 91, 109, 153, 172] [2, 7, 18, 66, 92, 110, 154, 173] [3, 11, 22, 70, 93, 111, 158, 174] [4, 9, 23, 71, 94, 112, 156, 175] [5, 10, 21, 69, 95, 113, 157, 176] [6, 14, 25, 73, 96, 114, 161, 177] [7, 12, 26, 74, 97, 115, 159, 178] [8, 13, 24, 72, 98, 116, 160, 179] [9, 17, 28, 76, 90, 99, 117, 164] [10, 15, 29, 77, 91, 100, 118, 162] [11, 16, 27, 75, 92, 101, 119, 163] [12, 20, 31, 79, 93, 102, 120, 167] [13, 18, 32, 80, 94, 103, 121, 165] [14, 19, 30, 78, 95, 104, 122, 166] [15, 23, 34, 82, 96, 105, 123, 170] [16, 21, 35, 83, 97, 106, 124, 168] [17, 22, 33, 81, 98, 107, 125, 169] [18, 26, 37, 85, 99, 108, 126, 173] [19, 24, 38, 86, 100, 109, 127, 171] [20, 25, 36, 84, 101, 110, 128, 172] [21, 29, 40, 88, 102, 111, 129, 176] [22, 27, 41, 89, 103, 112, 130, 174] [23, 28, 39, 87, 104, 113, 131, 175] [1, 24, 32, 43, 105, 114, 132, 179] [2, 25, 30, 44, 106, 115, 133, 177] [0, 26, 31, 42, 107, 116, 134, 178] [4, 27, 35, 46, 92, 108, 117, 135] [5, 28, 33, 47, 90, 109, 118, 136] [3, 29, 34, 45, 91, 110, 119, 137] [7, 30, 38, 49, 95, 111, 120, 138] [8, 31, 36, 50, 93, 112, 121, 139] [6, 32, 37, 48, 94, 113, 122, 140] [10, 33, 41, 52, 98, 114, 123, 141] [11, 34, 39, 53, 96, 115, 124, 142] [9, 35, 40, 51, 97, 116, 125, 143] [13, 36, 44, 55, 101, 117, 126, 144] [14, 37, 42, 56, 99, 118, 127, 145] [12, 38, 43, 54, 100, 119, 128, 146] [16, 39, 47, 58, 104, 120, 129, 147] [17, 40, 45, 59, 102, 121, 130, 148] [15, 41, 46, 57, 103, 122, 131, 149] [19, 42, 50, 61, 107, 123, 132, 150] [20, 43, 48, 62, 105, 124, 133, 151] [18, 44, 49, 60, 106, 125, 134, 152] [22, 45, 53, 64, 110, 126, 135, 153] [23, 46, 51, 65, 108, 127, 136, 154] [21, 47, 52, 63, 109, 128, 137, 155] [25, 48, 56, 67, 113, 129, 138, 156] [26, 49, 54, 68, 111, 130, 139, 157] [24, 50, 55, 66, 112, 131, 140, 158] [28, 51, 59, 70, 116, 132, 141, 159] [29, 52, 57, 71, 114, 133, 142, 160] [27, 53, 58, 69, 115, 134, 143, 161] [31, 54, 62, 73, 119, 135, 144, 162] [32, 55, 60, 74, 117, 136, 145, 163] [30, 56, 61, 72, 118, 137, 146, 164] [34, 57, 65, 76, 122, 138, 147, 165] [35, 58, 63, 77, 120, 139, 148, 166] [33, 59, 64, 75, 121, 140, 149, 167] [37, 60, 68, 79, 125, 141, 150, 168] [38, 61, 66, 80, 123, 142, 151, 169] [36, 62, 67, 78, 124, 143, 152, 170] [40, 63, 71, 82, 128, 144, 153, 171] [41, 64, 69, 83, 126, 145, 154, 172] [39, 65, 70, 81, 127, 146, 155, 173] [43, 66, 74, 85, 131, 147, 156, 174] [44, 67, 72, 86, 129, 148, 157, 175] [42, 68, 73, 84, 130, 149, 158, 176] [46, 69, 77, 88, 134, 150, 159, 177] [47, 70, 75, 89, 132, 151, 160, 178] [45, 71, 76, 87, 133, 152, 161, 179] [1, 49, 72, 80, 90, 137, 153, 162] [2, 50, 73, 78, 91, 135, 154, 163] [0, 48, 74, 79, 92, 136, 155, 164] [4, 52, 75, 83, 93, 140, 156, 165] [5, 53, 76, 81, 94, 138, 157, 166] [3, 51, 77, 82, 95, 139, 158, 167] [7, 55, 78, 86, 96, 143, 159, 168] [8, 56, 79, 84, 97, 141, 160, 169] [6, 54, 80, 85, 98, 142, 161, 170] [10, 58, 81, 89, 99, 146, 162, 171] [11, 59, 82, 87, 100, 144, 163, 172] [9, 57, 83, 88, 101, 145, 164, 173] [2, 13, 61, 84, 102, 149, 165, 174] [0, 14, 62, 85, 103, 147, 166, 175] [1, 12, 60, 86, 104, 148, 167, 176] [5, 16, 64, 87, 105, 152, 168, 177] [3, 17, 65, 88, 106, 150, 169, 178] [4, 15, 63, 89, 107, 151, 170, 179]
H_Z (90 checks, sparse supports)
[0, 9, 28, 72, 90, 116, 164, 175] [1, 10, 29, 73, 91, 114, 162, 176] [2, 11, 27, 74, 92, 115, 163, 174] [3, 12, 31, 75, 93, 119, 167, 178] [4, 13, 32, 76, 94, 117, 165, 179] [5, 14, 30, 77, 95, 118, 166, 177] [6, 15, 34, 78, 91, 96, 122, 170] [7, 16, 35, 79, 92, 97, 120, 168] [8, 17, 33, 80, 90, 98, 121, 169] [9, 18, 37, 81, 94, 99, 125, 173] [10, 19, 38, 82, 95, 100, 123, 171] [11, 20, 36, 83, 93, 101, 124, 172] [12, 21, 40, 84, 97, 102, 128, 176] [13, 22, 41, 85, 98, 103, 126, 174] [14, 23, 39, 86, 96, 104, 127, 175] [15, 24, 43, 87, 100, 105, 131, 179] [16, 25, 44, 88, 101, 106, 129, 177] [17, 26, 42, 89, 99, 107, 130, 178] [0, 18, 27, 46, 92, 103, 108, 134] [1, 19, 28, 47, 90, 104, 109, 132] [2, 20, 29, 45, 91, 102, 110, 133] [3, 21, 30, 49, 95, 106, 111, 137] [4, 22, 31, 50, 93, 107, 112, 135] [5, 23, 32, 48, 94, 105, 113, 136] [6, 24, 33, 52, 98, 109, 114, 140] [7, 25, 34, 53, 96, 110, 115, 138] [8, 26, 35, 51, 97, 108, 116, 139] [9, 27, 36, 55, 101, 112, 117, 143] [10, 28, 37, 56, 99, 113, 118, 141] [11, 29, 38, 54, 100, 111, 119, 142] [12, 30, 39, 58, 104, 115, 120, 146] [13, 31, 40, 59, 102, 116, 121, 144] [14, 32, 41, 57, 103, 114, 122, 145] [15, 33, 42, 61, 107, 118, 123, 149] [16, 34, 43, 62, 105, 119, 124, 147] [17, 35, 44, 60, 106, 117, 125, 148] [18, 36, 45, 64, 110, 121, 126, 152] [19, 37, 46, 65, 108, 122, 127, 150] [20, 38, 47, 63, 109, 120, 128, 151] [21, 39, 48, 67, 113, 124, 129, 155] [22, 40, 49, 68, 111, 125, 130, 153] [23, 41, 50, 66, 112, 123, 131, 154] [24, 42, 51, 70, 116, 127, 132, 158] [25, 43, 52, 71, 114, 128, 133, 156] [26, 44, 53, 69, 115, 126, 134, 157] [27, 45, 54, 73, 119, 130, 135, 161] [28, 46, 55, 74, 117, 131, 136, 159] [29, 47, 56, 72, 118, 129, 137, 160] [30, 48, 57, 76, 122, 133, 138, 164] [31, 49, 58, 77, 120, 134, 139, 162] [32, 50, 59, 75, 121, 132, 140, 163] [33, 51, 60, 79, 125, 136, 141, 167] [34, 52, 61, 80, 123, 137, 142, 165] [35, 53, 62, 78, 124, 135, 143, 166] [36, 54, 63, 82, 128, 139, 144, 170] [37, 55, 64, 83, 126, 140, 145, 168] [38, 56, 65, 81, 127, 138, 146, 169] [39, 57, 66, 85, 131, 142, 147, 173] [40, 58, 67, 86, 129, 143, 148, 171] [41, 59, 68, 84, 130, 141, 149, 172] [42, 60, 69, 88, 134, 145, 150, 176] [43, 61, 70, 89, 132, 146, 151, 174] [44, 62, 71, 87, 133, 144, 152, 175] [1, 45, 63, 72, 137, 148, 153, 179] [2, 46, 64, 73, 135, 149, 154, 177] [0, 47, 65, 74, 136, 147, 155, 178] [4, 48, 66, 75, 92, 140, 151, 156] [5, 49, 67, 76, 90, 138, 152, 157] [3, 50, 68, 77, 91, 139, 150, 158] [7, 51, 69, 78, 95, 143, 154, 159] [8, 52, 70, 79, 93, 141, 155, 160] [6, 53, 71, 80, 94, 142, 153, 161] [10, 54, 72, 81, 98, 146, 157, 162] [11, 55, 73, 82, 96, 144, 158, 163] [9, 56, 74, 83, 97, 145, 156, 164] [13, 57, 75, 84, 101, 149, 160, 165] [14, 58, 76, 85, 99, 147, 161, 166] [12, 59, 77, 86, 100, 148, 159, 167] [16, 60, 78, 87, 104, 152, 163, 168] [17, 61, 79, 88, 102, 150, 164, 169] [15, 62, 80, 89, 103, 151, 162, 170] [0, 19, 63, 81, 107, 155, 166, 171] [1, 20, 64, 82, 105, 153, 167, 172] [2, 18, 65, 83, 106, 154, 165, 173] [3, 22, 66, 84, 110, 158, 169, 174] [4, 23, 67, 85, 108, 156, 170, 175] [5, 21, 68, 86, 109, 157, 168, 176] [6, 25, 69, 87, 113, 161, 172, 177] [7, 26, 70, 88, 111, 159, 173, 178] [8, 24, 71, 89, 112, 160, 171, 179]
raw submission JSON