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[[186,14,18]] d ≤
n
186
k
14
d
18
kd²/n
24.387
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[16, 27, 36, 42, 45, 51, 67, 80, 85, 100, 101, 102, 111, 128, 139, 156, 166, 181]
d_Z 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[15, 16, 18, 28, 44, 54, 55, 56, 62, 70, 82, 83, 102, 133, 164, 171, 173, 180]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 6 · H_Z 6 (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 3–5 (mean 4.0) · H_Z 3–5 (mean 4.0)
trapping sets H_X (1,3)×93 (2,4)×279 (3,3)×93 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 93 (1,5): 93 (2,4): 279 (2,6): 1395 (2,8): 930 (3,3): 93 (3,5): 930 (3,7): 14322 (3,9): 31093 (3,11): 11439 (3,13): 930
trapping sets H_Z (1,3)×93 (2,4)×279 (3,3)×93 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 93 (1,5): 93 (2,4): 279 (2,6): 1395 (2,8): 930 (3,3): 93 (3,5): 930 (3,7): 14322 (3,9): 31093 (3,11): 11439 (3,13): 930

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction bivariate bicycle on Z_3 x Z_31: A = x0y0 + x2y5 + x1y28, B = x0y0 + x1y15 + x1y24 + x1y0 + x2y22
model Claude Claude Fable 5.1 (claimed, not verified)
date 2026-10-17
family bivariate 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

[[186,14,18]] — bivariate bicycle code on Z_3 x Z_31, check weight 8

Direction & hypothesis

Target: the unrestricted x weight-8 cell at n in 111..220, where the board's frontier for k >= 12 at d >= 8 is thin (frontier map of 2026-10-17). A (186, 14, 18) point at check weight 8 needed d >= 18 to clear every board entry with n' <= 186, k' >= 14, w' <= 8; the gate's novelty check reports it advances the cell on d, k, n (board-relative novelty from verify/validate_candidate.py; literature novelty unverified).

What was searched

Designed-common-zero bivariate bicycle sweep: for every torus Z_l x Z_m with l*m in 56..110 (n in 112..220), a point (alpha, beta) of mu_l' x mu_m' (l', m' the odd parts) with Frobenius orbit of size t >= 4 is fixed first and A, B are drawn only among the 3- and 5-term polynomials that vanish there (so k >= 2t by construction; this code's point is index [0, 5] in Z_3 x Z_31). Funnel for this job: 206923 candidates built, 154265 distinct with k >= 10 screened at 400 gf2_fast RIS trials, 67 promoted to 20,000 trials, 62 to 200,000 trials, 19 survived with the reading still above the board's weight-8 frontier (any reading below the needed d is final, since the search returns an upper bound).

Evidence trail

Witness-backed upper bound d <= 18: the lightest logical operator found is embedded in the submission (distance.X / distance.Z witnesses) and is checked by the verifier. Confirmation ladder on the box (gf2_fast, research/kit/surrogate.distance_rand_witness, 24 threads):

| trials per side | lightest logical found | |---|---| | 400 / 20,000 / 200,000 funnel readings | 18 | | 4,000,000 fast RIS trials per side, fresh seed (54222) | 18 (X) | | 4,000,000 fast RIS trials per side, fresh seed (62141) | 18 (X) | | 4,000,000 fast RIS trials per side, fresh seed (70060) | 18 (X) |

Ladder verdict: corroborated (min over fresh seeds = 18, claimed 18). The claim is an upper bound; no exactness is asserted.

Dead ends

  • Random (undesigned) bivariate bicycle supports in this band: 250,000 weight-6 draws gave only

    • 204 codes with k >= 10 and none above the frontier at 400 trials; 230,000 weight-8 draws gave 3,529 with k >= 10 and 5 survivors at 200,000 trials.

  • Random 2BGA supports on 150 non-abelian groups of order 56..110 (43,800 weight-6 draws): 146 with

    • k >= 10, none above the frontier at 400 trials.

  • Designed-common-zero weight-6 bicycle codes (45,176 draws, 7,306 with k >= 10): all readings at or

    • below the frontier; the [[150,16,8]] cell is reproduced many times but never beaten.

Tools

Model: Claude Fable 5.1 (Claude Code agent, campaign agent A03). Brev n2d-highcpu-64 box, gf2_fast RIS (make fast) with 60 threads; research/kit modules bb / group_algebra / coset / nonabelian_lp / gf2poly / search / surrogate / submit / prescreen; gate = verify/validate_candidate.py via research/kit/coordination.gate_and_record. Approximate compute: 17 CPU-box minutes for the sweep plus the ladder above.

Reproduction

bivariate bicycle on Z_3 x Z_31: A = x^0y^0 + x^2y^5 + x^1y^28, B = x^0y^0 + x^1y^15 + x^1y^24 + x^1y^0 + x^2y^22.

import sys; sys.path[:0] = ['research/kit', 'verify']
from bb import build_bb
HX, HZ = build_bb(3, 31, [(0, 0), (2, 5), (1, 28)], [(0, 0), (1, 15), (1, 24), (1, 0), (2, 22)])

Witness search: research/kit/surrogate.lightest_logical (8,000 NumPy trials, seeds in the witness_provenance block) tightened by gf2_fast RIS at 200,000 trials; ladder seeds are listed in the table above.

Parity checks

X-checks 93 (max weight 8) · Z-checks 93 (max weight 8)
H_X (93 checks, sparse supports)
[0, 59, 67, 93, 124, 139, 148, 177] [1, 60, 68, 94, 125, 140, 149, 178] [2, 61, 69, 95, 126, 141, 150, 179] [3, 31, 70, 96, 127, 142, 151, 180] [4, 32, 71, 97, 128, 143, 152, 181] [5, 33, 72, 98, 129, 144, 153, 182] [6, 34, 73, 99, 130, 145, 154, 183] [7, 35, 74, 100, 124, 131, 146, 184] [8, 36, 75, 101, 125, 132, 147, 185] [9, 37, 76, 102, 126, 133, 148, 155] [10, 38, 77, 103, 127, 134, 149, 156] [11, 39, 78, 104, 128, 135, 150, 157] [12, 40, 79, 105, 129, 136, 151, 158] [13, 41, 80, 106, 130, 137, 152, 159] [14, 42, 81, 107, 131, 138, 153, 160] [15, 43, 82, 108, 132, 139, 154, 161] [16, 44, 83, 109, 124, 133, 140, 162] [17, 45, 84, 110, 125, 134, 141, 163] [18, 46, 85, 111, 126, 135, 142, 164] [19, 47, 86, 112, 127, 136, 143, 165] [20, 48, 87, 113, 128, 137, 144, 166] [21, 49, 88, 114, 129, 138, 145, 167] [22, 50, 89, 115, 130, 139, 146, 168] [23, 51, 90, 116, 131, 140, 147, 169] [24, 52, 91, 117, 132, 141, 148, 170] [25, 53, 92, 118, 133, 142, 149, 171] [26, 54, 62, 119, 134, 143, 150, 172] [27, 55, 63, 120, 135, 144, 151, 173] [28, 56, 64, 121, 136, 145, 152, 174] [29, 57, 65, 122, 137, 146, 153, 175] [30, 58, 66, 123, 138, 147, 154, 176] [5, 31, 90, 115, 124, 155, 170, 179] [6, 32, 91, 116, 125, 156, 171, 180] [7, 33, 92, 117, 126, 157, 172, 181] [8, 34, 62, 118, 127, 158, 173, 182] [9, 35, 63, 119, 128, 159, 174, 183] [10, 36, 64, 120, 129, 160, 175, 184] [11, 37, 65, 121, 130, 161, 176, 185] [12, 38, 66, 122, 131, 155, 162, 177] [13, 39, 67, 123, 132, 156, 163, 178] [14, 40, 68, 93, 133, 157, 164, 179] [15, 41, 69, 94, 134, 158, 165, 180] [16, 42, 70, 95, 135, 159, 166, 181] [17, 43, 71, 96, 136, 160, 167, 182] [18, 44, 72, 97, 137, 161, 168, 183] [19, 45, 73, 98, 138, 162, 169, 184] [20, 46, 74, 99, 139, 163, 170, 185] [21, 47, 75, 100, 140, 155, 164, 171] [22, 48, 76, 101, 141, 156, 165, 172] [23, 49, 77, 102, 142, 157, 166, 173] [24, 50, 78, 103, 143, 158, 167, 174] [25, 51, 79, 104, 144, 159, 168, 175] [26, 52, 80, 105, 145, 160, 169, 176] [27, 53, 81, 106, 146, 161, 170, 177] [28, 54, 82, 107, 147, 162, 171, 178] [29, 55, 83, 108, 148, 163, 172, 179] [30, 56, 84, 109, 149, 164, 173, 180] [0, 57, 85, 110, 150, 165, 174, 181] [1, 58, 86, 111, 151, 166, 175, 182] [2, 59, 87, 112, 152, 167, 176, 183] [3, 60, 88, 113, 153, 168, 177, 184] [4, 61, 89, 114, 154, 169, 178, 185] [28, 36, 62, 93, 108, 117, 146, 155] [29, 37, 63, 94, 109, 118, 147, 156] [30, 38, 64, 95, 110, 119, 148, 157] [0, 39, 65, 96, 111, 120, 149, 158] [1, 40, 66, 97, 112, 121, 150, 159] [2, 41, 67, 98, 113, 122, 151, 160] [3, 42, 68, 99, 114, 123, 152, 161] [4, 43, 69, 93, 100, 115, 153, 162] [5, 44, 70, 94, 101, 116, 154, 163] [6, 45, 71, 95, 102, 117, 124, 164] [7, 46, 72, 96, 103, 118, 125, 165] [8, 47, 73, 97, 104, 119, 126, 166] [9, 48, 74, 98, 105, 120, 127, 167] [10, 49, 75, 99, 106, 121, 128, 168] [11, 50, 76, 100, 107, 122, 129, 169] [12, 51, 77, 101, 108, 123, 130, 170] [13, 52, 78, 93, 102, 109, 131, 171] [14, 53, 79, 94, 103, 110, 132, 172] [15, 54, 80, 95, 104, 111, 133, 173] [16, 55, 81, 96, 105, 112, 134, 174] [17, 56, 82, 97, 106, 113, 135, 175] [18, 57, 83, 98, 107, 114, 136, 176] [19, 58, 84, 99, 108, 115, 137, 177] [20, 59, 85, 100, 109, 116, 138, 178] [21, 60, 86, 101, 110, 117, 139, 179] [22, 61, 87, 102, 111, 118, 140, 180] [23, 31, 88, 103, 112, 119, 141, 181] [24, 32, 89, 104, 113, 120, 142, 182] [25, 33, 90, 105, 114, 121, 143, 183] [26, 34, 91, 106, 115, 122, 144, 184] [27, 35, 92, 107, 116, 123, 145, 185]
H_Z (93 checks, sparse supports)
[0, 40, 62, 69, 78, 93, 150, 158] [1, 41, 63, 70, 79, 94, 151, 159] [2, 42, 64, 71, 80, 95, 152, 160] [3, 43, 65, 72, 81, 96, 153, 161] [4, 44, 66, 73, 82, 97, 154, 162] [5, 45, 67, 74, 83, 98, 124, 163] [6, 46, 68, 75, 84, 99, 125, 164] [7, 47, 69, 76, 85, 100, 126, 165] [8, 48, 70, 77, 86, 101, 127, 166] [9, 49, 71, 78, 87, 102, 128, 167] [10, 50, 72, 79, 88, 103, 129, 168] [11, 51, 73, 80, 89, 104, 130, 169] [12, 52, 74, 81, 90, 105, 131, 170] [13, 53, 75, 82, 91, 106, 132, 171] [14, 54, 76, 83, 92, 107, 133, 172] [15, 55, 62, 77, 84, 108, 134, 173] [16, 56, 63, 78, 85, 109, 135, 174] [17, 57, 64, 79, 86, 110, 136, 175] [18, 58, 65, 80, 87, 111, 137, 176] [19, 59, 66, 81, 88, 112, 138, 177] [20, 60, 67, 82, 89, 113, 139, 178] [21, 61, 68, 83, 90, 114, 140, 179] [22, 31, 69, 84, 91, 115, 141, 180] [23, 32, 70, 85, 92, 116, 142, 181] [24, 33, 62, 71, 86, 117, 143, 182] [25, 34, 63, 72, 87, 118, 144, 183] [26, 35, 64, 73, 88, 119, 145, 184] [27, 36, 65, 74, 89, 120, 146, 185] [28, 37, 66, 75, 90, 121, 147, 155] [29, 38, 67, 76, 91, 122, 148, 156] [30, 39, 68, 77, 92, 123, 149, 157] [0, 7, 16, 31, 71, 96, 124, 181] [1, 8, 17, 32, 72, 97, 125, 182] [2, 9, 18, 33, 73, 98, 126, 183] [3, 10, 19, 34, 74, 99, 127, 184] [4, 11, 20, 35, 75, 100, 128, 185] [5, 12, 21, 36, 76, 101, 129, 155] [6, 13, 22, 37, 77, 102, 130, 156] [7, 14, 23, 38, 78, 103, 131, 157] [8, 15, 24, 39, 79, 104, 132, 158] [9, 16, 25, 40, 80, 105, 133, 159] [10, 17, 26, 41, 81, 106, 134, 160] [11, 18, 27, 42, 82, 107, 135, 161] [12, 19, 28, 43, 83, 108, 136, 162] [13, 20, 29, 44, 84, 109, 137, 163] [14, 21, 30, 45, 85, 110, 138, 164] [0, 15, 22, 46, 86, 111, 139, 165] [1, 16, 23, 47, 87, 112, 140, 166] [2, 17, 24, 48, 88, 113, 141, 167] [3, 18, 25, 49, 89, 114, 142, 168] [4, 19, 26, 50, 90, 115, 143, 169] [5, 20, 27, 51, 91, 116, 144, 170] [6, 21, 28, 52, 92, 117, 145, 171] [7, 22, 29, 53, 62, 118, 146, 172] [8, 23, 30, 54, 63, 119, 147, 173] [0, 9, 24, 55, 64, 120, 148, 174] [1, 10, 25, 56, 65, 121, 149, 175] [2, 11, 26, 57, 66, 122, 150, 176] [3, 12, 27, 58, 67, 123, 151, 177] [4, 13, 28, 59, 68, 93, 152, 178] [5, 14, 29, 60, 69, 94, 153, 179] [6, 15, 30, 61, 70, 95, 154, 180] [9, 31, 38, 47, 62, 119, 127, 155] [10, 32, 39, 48, 63, 120, 128, 156] [11, 33, 40, 49, 64, 121, 129, 157] [12, 34, 41, 50, 65, 122, 130, 158] [13, 35, 42, 51, 66, 123, 131, 159] [14, 36, 43, 52, 67, 93, 132, 160] [15, 37, 44, 53, 68, 94, 133, 161] [16, 38, 45, 54, 69, 95, 134, 162] [17, 39, 46, 55, 70, 96, 135, 163] [18, 40, 47, 56, 71, 97, 136, 164] [19, 41, 48, 57, 72, 98, 137, 165] [20, 42, 49, 58, 73, 99, 138, 166] [21, 43, 50, 59, 74, 100, 139, 167] [22, 44, 51, 60, 75, 101, 140, 168] [23, 45, 52, 61, 76, 102, 141, 169] [24, 31, 46, 53, 77, 103, 142, 170] [25, 32, 47, 54, 78, 104, 143, 171] [26, 33, 48, 55, 79, 105, 144, 172] [27, 34, 49, 56, 80, 106, 145, 173] [28, 35, 50, 57, 81, 107, 146, 174] [29, 36, 51, 58, 82, 108, 147, 175] [30, 37, 52, 59, 83, 109, 148, 176] [0, 38, 53, 60, 84, 110, 149, 177] [1, 39, 54, 61, 85, 111, 150, 178] [2, 31, 40, 55, 86, 112, 151, 179] [3, 32, 41, 56, 87, 113, 152, 180] [4, 33, 42, 57, 88, 114, 153, 181] [5, 34, 43, 58, 89, 115, 154, 182] [6, 35, 44, 59, 90, 116, 124, 183] [7, 36, 45, 60, 91, 117, 125, 184] [8, 37, 46, 61, 92, 118, 126, 185]
Code ID 186-14-18 · download JSON · raw on GitHub