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[[204,22,17]] d ≤
n
204
k
22
d
17
kd²/n
31.167
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · 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 17 · witness weight 17 (claimed upper_bound)
witness operator (support, 17 qubits)
[25, 29, 39, 46, 68, 74, 75, 78, 111, 126, 138, 141, 145, 160, 191, 197, 198]
d_Z 17 · witness weight 17 (claimed upper_bound)
witness operator (support, 17 qubits)
[5, 6, 12, 43, 58, 62, 65, 77, 92, 125, 128, 129, 135, 157, 164, 174, 178]
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 4 · H_Z 4
trapping sets H_X (1,4)×204 (2,6)×2856 (3,6)×1836 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 204 (2,6): 2856 (3,6): 1836 (3,8): 54468 (3,10): 5712
trapping sets H_Z (1,4)×204 (2,6)×2856 (3,6)×1836 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 204 (2,6): 2856 (3,6): 1836 (3,8): 54468 (3,10): 5712

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction bivariate bicycle on Z_3 x Z_34: A = x0y0 + x2y19 + x2y31 + x0y16, B = x0y0 + x1y22 + x1y26 + x0y31
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

[[204,22,17]] — bivariate bicycle code on Z_3 x Z_34, 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 (204, 22, 17) point at check weight 8 needed d >= 16 to clear every board entry with n' <= 204, k' >= 22, 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 4- and 4-term polynomials that vanish there (so k >= 2t by construction; this code's point is index [1, 1] in Z_3 x Z_17). Funnel for this job: 30826 candidates built, 20444 distinct with k >= 10 screened at 400 gf2_fast RIS trials, 205 promoted to 20,000 trials, 179 to 200,000 trials, 179 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 <= 17: 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, 30 threads):

| trials per side | lightest logical found | |---|---| | 400 / 20,000 / 200,000 funnel readings | 17 | | 10,000,000 fast RIS trials per side, fresh seed (47713) | 17 (X) | | 10,000,000 fast RIS trials per side, fresh seed (55632) | 17 (X) | | 10,000,000 fast RIS trials per side, fresh seed (63551) | 17 (X) |

Ladder verdict: corroborated (min over fresh seeds = 17, claimed 17). 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: 5 CPU-box minutes for the sweep plus the ladder above.

Reproduction

bivariate bicycle on Z_3 x Z_34: A = x^0y^0 + x^2y^19 + x^2y^31 + x^0y^16, B = x^0y^0 + x^1y^22 + x^1y^26 + x^0y^31.

import sys; sys.path[:0] = ['research/kit', 'verify']
from bb import build_bb
HX, HZ = build_bb(3, 34, [(0, 0), (2, 19), (2, 31), (0, 16)], [(0, 0), (1, 22), (1, 26), (0, 31)])

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 102 (max weight 8) · Z-checks 102 (max weight 8)
H_X (102 checks, sparse supports)
[0, 16, 87, 99, 102, 133, 158, 162] [1, 17, 88, 100, 103, 134, 159, 163] [2, 18, 89, 101, 104, 135, 160, 164] [3, 19, 68, 90, 102, 105, 161, 165] [4, 20, 69, 91, 103, 106, 162, 166] [5, 21, 70, 92, 104, 107, 163, 167] [6, 22, 71, 93, 105, 108, 164, 168] [7, 23, 72, 94, 106, 109, 165, 169] [8, 24, 73, 95, 107, 110, 136, 166] [9, 25, 74, 96, 108, 111, 137, 167] [10, 26, 75, 97, 109, 112, 138, 168] [11, 27, 76, 98, 110, 113, 139, 169] [12, 28, 77, 99, 111, 114, 136, 140] [13, 29, 78, 100, 112, 115, 137, 141] [14, 30, 79, 101, 113, 116, 138, 142] [15, 31, 68, 80, 114, 117, 139, 143] [16, 32, 69, 81, 115, 118, 140, 144] [17, 33, 70, 82, 116, 119, 141, 145] [0, 18, 71, 83, 117, 120, 142, 146] [1, 19, 72, 84, 118, 121, 143, 147] [2, 20, 73, 85, 119, 122, 144, 148] [3, 21, 74, 86, 120, 123, 145, 149] [4, 22, 75, 87, 121, 124, 146, 150] [5, 23, 76, 88, 122, 125, 147, 151] [6, 24, 77, 89, 123, 126, 148, 152] [7, 25, 78, 90, 124, 127, 149, 153] [8, 26, 79, 91, 125, 128, 150, 154] [9, 27, 80, 92, 126, 129, 151, 155] [10, 28, 81, 93, 127, 130, 152, 156] [11, 29, 82, 94, 128, 131, 153, 157] [12, 30, 83, 95, 129, 132, 154, 158] [13, 31, 84, 96, 130, 133, 155, 159] [14, 32, 85, 97, 131, 134, 156, 160] [15, 33, 86, 98, 132, 135, 157, 161] [19, 31, 34, 50, 136, 167, 192, 196] [20, 32, 35, 51, 137, 168, 193, 197] [21, 33, 36, 52, 138, 169, 194, 198] [0, 22, 37, 53, 136, 139, 195, 199] [1, 23, 38, 54, 137, 140, 196, 200] [2, 24, 39, 55, 138, 141, 197, 201] [3, 25, 40, 56, 139, 142, 198, 202] [4, 26, 41, 57, 140, 143, 199, 203] [5, 27, 42, 58, 141, 144, 170, 200] [6, 28, 43, 59, 142, 145, 171, 201] [7, 29, 44, 60, 143, 146, 172, 202] [8, 30, 45, 61, 144, 147, 173, 203] [9, 31, 46, 62, 145, 148, 170, 174] [10, 32, 47, 63, 146, 149, 171, 175] [11, 33, 48, 64, 147, 150, 172, 176] [0, 12, 49, 65, 148, 151, 173, 177] [1, 13, 50, 66, 149, 152, 174, 178] [2, 14, 51, 67, 150, 153, 175, 179] [3, 15, 34, 52, 151, 154, 176, 180] [4, 16, 35, 53, 152, 155, 177, 181] [5, 17, 36, 54, 153, 156, 178, 182] [6, 18, 37, 55, 154, 157, 179, 183] [7, 19, 38, 56, 155, 158, 180, 184] [8, 20, 39, 57, 156, 159, 181, 185] [9, 21, 40, 58, 157, 160, 182, 186] [10, 22, 41, 59, 158, 161, 183, 187] [11, 23, 42, 60, 159, 162, 184, 188] [12, 24, 43, 61, 160, 163, 185, 189] [13, 25, 44, 62, 161, 164, 186, 190] [14, 26, 45, 63, 162, 165, 187, 191] [15, 27, 46, 64, 163, 166, 188, 192] [16, 28, 47, 65, 164, 167, 189, 193] [17, 29, 48, 66, 165, 168, 190, 194] [18, 30, 49, 67, 166, 169, 191, 195] [53, 65, 68, 84, 124, 128, 170, 201] [54, 66, 69, 85, 125, 129, 171, 202] [55, 67, 70, 86, 126, 130, 172, 203] [34, 56, 71, 87, 127, 131, 170, 173] [35, 57, 72, 88, 128, 132, 171, 174] [36, 58, 73, 89, 129, 133, 172, 175] [37, 59, 74, 90, 130, 134, 173, 176] [38, 60, 75, 91, 131, 135, 174, 177] [39, 61, 76, 92, 102, 132, 175, 178] [40, 62, 77, 93, 103, 133, 176, 179] [41, 63, 78, 94, 104, 134, 177, 180] [42, 64, 79, 95, 105, 135, 178, 181] [43, 65, 80, 96, 102, 106, 179, 182] [44, 66, 81, 97, 103, 107, 180, 183] [45, 67, 82, 98, 104, 108, 181, 184] [34, 46, 83, 99, 105, 109, 182, 185] [35, 47, 84, 100, 106, 110, 183, 186] [36, 48, 85, 101, 107, 111, 184, 187] [37, 49, 68, 86, 108, 112, 185, 188] [38, 50, 69, 87, 109, 113, 186, 189] [39, 51, 70, 88, 110, 114, 187, 190] [40, 52, 71, 89, 111, 115, 188, 191] [41, 53, 72, 90, 112, 116, 189, 192] [42, 54, 73, 91, 113, 117, 190, 193] [43, 55, 74, 92, 114, 118, 191, 194] [44, 56, 75, 93, 115, 119, 192, 195] [45, 57, 76, 94, 116, 120, 193, 196] [46, 58, 77, 95, 117, 121, 194, 197] [47, 59, 78, 96, 118, 122, 195, 198] [48, 60, 79, 97, 119, 123, 196, 199] [49, 61, 80, 98, 120, 124, 197, 200] [50, 62, 81, 99, 121, 125, 198, 201] [51, 63, 82, 100, 122, 126, 199, 202] [52, 64, 83, 101, 123, 127, 200, 203]
H_Z (102 checks, sparse supports)
[0, 3, 76, 80, 102, 120, 139, 151] [1, 4, 77, 81, 103, 121, 140, 152] [2, 5, 78, 82, 104, 122, 141, 153] [3, 6, 79, 83, 105, 123, 142, 154] [4, 7, 80, 84, 106, 124, 143, 155] [5, 8, 81, 85, 107, 125, 144, 156] [6, 9, 82, 86, 108, 126, 145, 157] [7, 10, 83, 87, 109, 127, 146, 158] [8, 11, 84, 88, 110, 128, 147, 159] [9, 12, 85, 89, 111, 129, 148, 160] [10, 13, 86, 90, 112, 130, 149, 161] [11, 14, 87, 91, 113, 131, 150, 162] [12, 15, 88, 92, 114, 132, 151, 163] [13, 16, 89, 93, 115, 133, 152, 164] [14, 17, 90, 94, 116, 134, 153, 165] [15, 18, 91, 95, 117, 135, 154, 166] [16, 19, 92, 96, 102, 118, 155, 167] [17, 20, 93, 97, 103, 119, 156, 168] [18, 21, 94, 98, 104, 120, 157, 169] [19, 22, 95, 99, 105, 121, 136, 158] [20, 23, 96, 100, 106, 122, 137, 159] [21, 24, 97, 101, 107, 123, 138, 160] [22, 25, 68, 98, 108, 124, 139, 161] [23, 26, 69, 99, 109, 125, 140, 162] [24, 27, 70, 100, 110, 126, 141, 163] [25, 28, 71, 101, 111, 127, 142, 164] [26, 29, 68, 72, 112, 128, 143, 165] [27, 30, 69, 73, 113, 129, 144, 166] [28, 31, 70, 74, 114, 130, 145, 167] [29, 32, 71, 75, 115, 131, 146, 168] [30, 33, 72, 76, 116, 132, 147, 169] [0, 31, 73, 77, 117, 133, 136, 148] [1, 32, 74, 78, 118, 134, 137, 149] [2, 33, 75, 79, 119, 135, 138, 150] [8, 12, 34, 37, 136, 154, 173, 185] [9, 13, 35, 38, 137, 155, 174, 186] [10, 14, 36, 39, 138, 156, 175, 187] [11, 15, 37, 40, 139, 157, 176, 188] [12, 16, 38, 41, 140, 158, 177, 189] [13, 17, 39, 42, 141, 159, 178, 190] [14, 18, 40, 43, 142, 160, 179, 191] [15, 19, 41, 44, 143, 161, 180, 192] [16, 20, 42, 45, 144, 162, 181, 193] [17, 21, 43, 46, 145, 163, 182, 194] [18, 22, 44, 47, 146, 164, 183, 195] [19, 23, 45, 48, 147, 165, 184, 196] [20, 24, 46, 49, 148, 166, 185, 197] [21, 25, 47, 50, 149, 167, 186, 198] [22, 26, 48, 51, 150, 168, 187, 199] [23, 27, 49, 52, 151, 169, 188, 200] [24, 28, 50, 53, 136, 152, 189, 201] [25, 29, 51, 54, 137, 153, 190, 202] [26, 30, 52, 55, 138, 154, 191, 203] [27, 31, 53, 56, 139, 155, 170, 192] [28, 32, 54, 57, 140, 156, 171, 193] [29, 33, 55, 58, 141, 157, 172, 194] [0, 30, 56, 59, 142, 158, 173, 195] [1, 31, 57, 60, 143, 159, 174, 196] [2, 32, 58, 61, 144, 160, 175, 197] [3, 33, 59, 62, 145, 161, 176, 198] [0, 4, 60, 63, 146, 162, 177, 199] [1, 5, 61, 64, 147, 163, 178, 200] [2, 6, 62, 65, 148, 164, 179, 201] [3, 7, 63, 66, 149, 165, 180, 202] [4, 8, 64, 67, 150, 166, 181, 203] [5, 9, 34, 65, 151, 167, 170, 182] [6, 10, 35, 66, 152, 168, 171, 183] [7, 11, 36, 67, 153, 169, 172, 184] [42, 46, 68, 71, 105, 117, 170, 188] [43, 47, 69, 72, 106, 118, 171, 189] [44, 48, 70, 73, 107, 119, 172, 190] [45, 49, 71, 74, 108, 120, 173, 191] [46, 50, 72, 75, 109, 121, 174, 192] [47, 51, 73, 76, 110, 122, 175, 193] [48, 52, 74, 77, 111, 123, 176, 194] [49, 53, 75, 78, 112, 124, 177, 195] [50, 54, 76, 79, 113, 125, 178, 196] [51, 55, 77, 80, 114, 126, 179, 197] [52, 56, 78, 81, 115, 127, 180, 198] [53, 57, 79, 82, 116, 128, 181, 199] [54, 58, 80, 83, 117, 129, 182, 200] [55, 59, 81, 84, 118, 130, 183, 201] [56, 60, 82, 85, 119, 131, 184, 202] [57, 61, 83, 86, 120, 132, 185, 203] [58, 62, 84, 87, 121, 133, 170, 186] [59, 63, 85, 88, 122, 134, 171, 187] [60, 64, 86, 89, 123, 135, 172, 188] [61, 65, 87, 90, 102, 124, 173, 189] [62, 66, 88, 91, 103, 125, 174, 190] [63, 67, 89, 92, 104, 126, 175, 191] [34, 64, 90, 93, 105, 127, 176, 192] [35, 65, 91, 94, 106, 128, 177, 193] [36, 66, 92, 95, 107, 129, 178, 194] [37, 67, 93, 96, 108, 130, 179, 195] [34, 38, 94, 97, 109, 131, 180, 196] [35, 39, 95, 98, 110, 132, 181, 197] [36, 40, 96, 99, 111, 133, 182, 198] [37, 41, 97, 100, 112, 134, 183, 199] [38, 42, 98, 101, 113, 135, 184, 200] [39, 43, 68, 99, 102, 114, 185, 201] [40, 44, 69, 100, 103, 115, 186, 202] [41, 45, 70, 101, 104, 116, 187, 203]
Code ID 204-22-17 · download JSON · raw on GitHub