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[[186,20,16]] d ≤
n
186
k
20
d
16
kd²/n
27.527
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[5, 9, 27, 37, 46, 73, 95, 98, 100, 101, 103, 116, 121, 128, 130, 152]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[31, 33, 38, 39, 46, 73, 75, 86, 88, 91, 98, 130, 139, 143, 148, 184]
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): 14229 (3,9): 30504 (3,11): 14043 (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): 14229 (3,9): 30504 (3,11): 14043 (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 + x0y5 + x0y7, B = x0y0 + x2y29 + x2y2 + x0y18 + x1y19
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

[[186,20,16]] — 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, 20, 16) point at check weight 8 needed d >= 16 to clear every board entry with n' <= 186, k' >= 20, 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 [1, 11] in Z_3 x Z_31). Funnel for this job: 73000 candidates built, 54547 distinct with k >= 10 screened at 400 gf2_fast RIS trials, 25 promoted to 20,000 trials, 21 to 200,000 trials, 7 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 <= 16: 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 | 16 | | 4,000,000 fast RIS trials per side, fresh seed (52596) | 16 (X) | | 4,000,000 fast RIS trials per side, fresh seed (60515) | 16 (X) | | 4,000,000 fast RIS trials per side, fresh seed (68434) | 16 (X) |

Ladder verdict: corroborated (min over fresh seeds = 16, claimed 16). 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: 8 CPU-box minutes for the sweep plus the ladder above.

Reproduction

bivariate bicycle on Z_3 x Z_31: A = x^0y^0 + x^0y^5 + x^0y^7, B = x^0y^0 + x^2y^29 + x^2y^2 + x^0y^18 + x^1y^19.

import sys; sys.path[:0] = ['research/kit', 'verify']
from bb import build_bb
HX, HZ = build_bb(3, 31, [(0, 0), (0, 5), (0, 7)], [(0, 0), (2, 29), (2, 2), (0, 18), (1, 19)])

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