[[210,10,21]] · QEC Challenge

[[210,10,21]] d ≤

210

kd²/n

21.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 21, d_Z ≤ 21 · 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 21 · witness weight 21 (claimed upper_bound)

witness operator (support, 21 qubits)

[15, 20, 21, 36, 43, 48, 56, 76, 85, 88, 97, 108, 110, 121, 129, 145, 146, 162, 190, 197, 201]

d_Z 21 · witness weight 21 (claimed upper_bound)

witness operator (support, 21 qubits)

[9, 11, 12, 13, 27, 36, 45, 60, 63, 64, 80, 104, 106, 117, 135, 137, 141, 164, 178, 183, 190]

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)×210 (2,6)×2940 (3,6)×1575 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,4): 210 (2,6): 2940 (3,6): 1575 (3,8): 57015 (3,10): 5880

trapping sets H_Z (1,4)×210 (2,6)×2940 (3,6)×1575 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,4): 210 (2,6): 2940 (3,6): 1575 (3,8): 57015 (3,10): 5880

Construction & provenance

authors @FarLab

provenance submitted through the challenge

novelty novelty not audited

construction bivariate bicycle on Z_3 x Z_35: A = x⁰y⁰ + x⁰y¹⁶ + x²y¹⁴ + x¹y¹, B = x⁰y⁰ + x⁰y¹² + x²y² + x¹y¹⁸

model 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

[[210,10,21]] — bivariate bicycle code on Z_3 x Z_35, 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 (210, 10, 21) point at check weight 8 needed d >= 21 to clear every board entry with n' <= 210, k' >= 10, 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, 7] in Z_3 x Z_35). 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 <= 21: 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 | 21 | | 10,000,000 fast RIS trials per side, fresh seed (51102) | 21 (X) | | 10,000,000 fast RIS trials per side, fresh seed (59021) | 21 (X) | | 10,000,000 fast RIS trials per side, fresh seed (66940) | 21 (X) |

Ladder verdict: corroborated (min over fresh seeds = 21, claimed 21). 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_35: A = x^0y^0 + x^0y^16 + x^2y^14 + x^1y^1, B = x^0y^0 + x^0y^12 + x^2y^2 + x^1y^18.

import sys; sys.path[:0] = ['research/kit', 'verify']
from bb import build_bb
HX, HZ = build_bb(3, 35, [(0, 0), (0, 16), (2, 14), (1, 1)], [(0, 0), (0, 12), (2, 2), (1, 18)])

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 105 (max weight 8) · Z-checks 105 (max weight 8)

H_X (105 checks, sparse supports)

[0, 16, 36, 84, 105, 117, 158, 177] [1, 17, 37, 85, 106, 118, 159, 178] [2, 18, 38, 86, 107, 119, 160, 179] [3, 19, 39, 87, 108, 120, 161, 180] [4, 20, 40, 88, 109, 121, 162, 181] [5, 21, 41, 89, 110, 122, 163, 182] [6, 22, 42, 90, 111, 123, 164, 183] [7, 23, 43, 91, 112, 124, 165, 184] [8, 24, 44, 92, 113, 125, 166, 185] [9, 25, 45, 93, 114, 126, 167, 186] [10, 26, 46, 94, 115, 127, 168, 187] [11, 27, 47, 95, 116, 128, 169, 188] [12, 28, 48, 96, 117, 129, 170, 189] [13, 29, 49, 97, 118, 130, 171, 190] [14, 30, 50, 98, 119, 131, 172, 191] [15, 31, 51, 99, 120, 132, 173, 192] [16, 32, 52, 100, 121, 133, 174, 193] [17, 33, 53, 101, 122, 134, 140, 194] [18, 34, 54, 102, 123, 135, 141, 195] [0, 19, 55, 103, 124, 136, 142, 196] [1, 20, 56, 104, 125, 137, 143, 197] [2, 21, 57, 70, 126, 138, 144, 198] [3, 22, 58, 71, 127, 139, 145, 199] [4, 23, 59, 72, 105, 128, 146, 200] [5, 24, 60, 73, 106, 129, 147, 201] [6, 25, 61, 74, 107, 130, 148, 202] [7, 26, 62, 75, 108, 131, 149, 203] [8, 27, 63, 76, 109, 132, 150, 204] [9, 28, 64, 77, 110, 133, 151, 205] [10, 29, 65, 78, 111, 134, 152, 206] [11, 30, 66, 79, 112, 135, 153, 207] [12, 31, 67, 80, 113, 136, 154, 208] [13, 32, 68, 81, 114, 137, 155, 209] [14, 33, 69, 82, 115, 138, 156, 175] [15, 34, 35, 83, 116, 139, 157, 176] [14, 35, 51, 71, 107, 140, 152, 193] [15, 36, 52, 72, 108, 141, 153, 194] [16, 37, 53, 73, 109, 142, 154, 195] [17, 38, 54, 74, 110, 143, 155, 196] [18, 39, 55, 75, 111, 144, 156, 197] [19, 40, 56, 76, 112, 145, 157, 198] [20, 41, 57, 77, 113, 146, 158, 199] [21, 42, 58, 78, 114, 147, 159, 200] [22, 43, 59, 79, 115, 148, 160, 201] [23, 44, 60, 80, 116, 149, 161, 202] [24, 45, 61, 81, 117, 150, 162, 203] [25, 46, 62, 82, 118, 151, 163, 204] [26, 47, 63, 83, 119, 152, 164, 205] [27, 48, 64, 84, 120, 153, 165, 206] [28, 49, 65, 85, 121, 154, 166, 207] [29, 50, 66, 86, 122, 155, 167, 208] [30, 51, 67, 87, 123, 156, 168, 209] [31, 52, 68, 88, 124, 157, 169, 175] [32, 53, 69, 89, 125, 158, 170, 176] [33, 35, 54, 90, 126, 159, 171, 177] [34, 36, 55, 91, 127, 160, 172, 178] [0, 37, 56, 92, 128, 161, 173, 179] [1, 38, 57, 93, 129, 162, 174, 180] [2, 39, 58, 94, 130, 140, 163, 181] [3, 40, 59, 95, 131, 141, 164, 182] [4, 41, 60, 96, 132, 142, 165, 183] [5, 42, 61, 97, 133, 143, 166, 184] [6, 43, 62, 98, 134, 144, 167, 185] [7, 44, 63, 99, 135, 145, 168, 186] [8, 45, 64, 100, 136, 146, 169, 187] [9, 46, 65, 101, 137, 147, 170, 188] [10, 47, 66, 102, 138, 148, 171, 189] [11, 48, 67, 103, 139, 149, 172, 190] [12, 49, 68, 104, 105, 150, 173, 191] [13, 50, 69, 70, 106, 151, 174, 192] [1, 49, 70, 86, 123, 142, 175, 187] [2, 50, 71, 87, 124, 143, 176, 188] [3, 51, 72, 88, 125, 144, 177, 189] [4, 52, 73, 89, 126, 145, 178, 190] [5, 53, 74, 90, 127, 146, 179, 191] [6, 54, 75, 91, 128, 147, 180, 192] [7, 55, 76, 92, 129, 148, 181, 193] [8, 56, 77, 93, 130, 149, 182, 194] [9, 57, 78, 94, 131, 150, 183, 195] [10, 58, 79, 95, 132, 151, 184, 196] [11, 59, 80, 96, 133, 152, 185, 197] [12, 60, 81, 97, 134, 153, 186, 198] [13, 61, 82, 98, 135, 154, 187, 199] [14, 62, 83, 99, 136, 155, 188, 200] [15, 63, 84, 100, 137, 156, 189, 201] [16, 64, 85, 101, 138, 157, 190, 202] [17, 65, 86, 102, 139, 158, 191, 203] [18, 66, 87, 103, 105, 159, 192, 204] [19, 67, 88, 104, 106, 160, 193, 205] [20, 68, 70, 89, 107, 161, 194, 206] [21, 69, 71, 90, 108, 162, 195, 207] [22, 35, 72, 91, 109, 163, 196, 208] [23, 36, 73, 92, 110, 164, 197, 209] [24, 37, 74, 93, 111, 165, 175, 198] [25, 38, 75, 94, 112, 166, 176, 199] [26, 39, 76, 95, 113, 167, 177, 200] [27, 40, 77, 96, 114, 168, 178, 201] [28, 41, 78, 97, 115, 169, 179, 202] [29, 42, 79, 98, 116, 170, 180, 203] [30, 43, 80, 99, 117, 171, 181, 204] [31, 44, 81, 100, 118, 172, 182, 205] [32, 45, 82, 101, 119, 173, 183, 206] [33, 46, 83, 102, 120, 174, 184, 207] [34, 47, 84, 103, 121, 140, 185, 208] [0, 48, 85, 104, 122, 141, 186, 209]

H_Z (105 checks, sparse supports)

[0, 23, 68, 87, 105, 124, 161, 209] [1, 24, 69, 88, 106, 125, 162, 175] [2, 25, 35, 89, 107, 126, 163, 176] [3, 26, 36, 90, 108, 127, 164, 177] [4, 27, 37, 91, 109, 128, 165, 178] [5, 28, 38, 92, 110, 129, 166, 179] [6, 29, 39, 93, 111, 130, 167, 180] [7, 30, 40, 94, 112, 131, 168, 181] [8, 31, 41, 95, 113, 132, 169, 182] [9, 32, 42, 96, 114, 133, 170, 183] [10, 33, 43, 97, 115, 134, 171, 184] [11, 34, 44, 98, 116, 135, 172, 185] [0, 12, 45, 99, 117, 136, 173, 186] [1, 13, 46, 100, 118, 137, 174, 187] [2, 14, 47, 101, 119, 138, 140, 188] [3, 15, 48, 102, 120, 139, 141, 189] [4, 16, 49, 103, 105, 121, 142, 190] [5, 17, 50, 104, 106, 122, 143, 191] [6, 18, 51, 70, 107, 123, 144, 192] [7, 19, 52, 71, 108, 124, 145, 193] [8, 20, 53, 72, 109, 125, 146, 194] [9, 21, 54, 73, 110, 126, 147, 195] [10, 22, 55, 74, 111, 127, 148, 196] [11, 23, 56, 75, 112, 128, 149, 197] [12, 24, 57, 76, 113, 129, 150, 198] [13, 25, 58, 77, 114, 130, 151, 199] [14, 26, 59, 78, 115, 131, 152, 200] [15, 27, 60, 79, 116, 132, 153, 201] [16, 28, 61, 80, 117, 133, 154, 202] [17, 29, 62, 81, 118, 134, 155, 203] [18, 30, 63, 82, 119, 135, 156, 204] [19, 31, 64, 83, 120, 136, 157, 205] [20, 32, 65, 84, 121, 137, 158, 206] [21, 33, 66, 85, 122, 138, 159, 207] [22, 34, 67, 86, 123, 139, 160, 208] [17, 35, 58, 103, 139, 140, 159, 196] [18, 36, 59, 104, 105, 141, 160, 197] [19, 37, 60, 70, 106, 142, 161, 198] [20, 38, 61, 71, 107, 143, 162, 199] [21, 39, 62, 72, 108, 144, 163, 200] [22, 40, 63, 73, 109, 145, 164, 201] [23, 41, 64, 74, 110, 146, 165, 202] [24, 42, 65, 75, 111, 147, 166, 203] [25, 43, 66, 76, 112, 148, 167, 204] [26, 44, 67, 77, 113, 149, 168, 205] [27, 45, 68, 78, 114, 150, 169, 206] [28, 46, 69, 79, 115, 151, 170, 207] [29, 35, 47, 80, 116, 152, 171, 208] [30, 36, 48, 81, 117, 153, 172, 209] [31, 37, 49, 82, 118, 154, 173, 175] [32, 38, 50, 83, 119, 155, 174, 176] [33, 39, 51, 84, 120, 140, 156, 177] [34, 40, 52, 85, 121, 141, 157, 178] [0, 41, 53, 86, 122, 142, 158, 179] [1, 42, 54, 87, 123, 143, 159, 180] [2, 43, 55, 88, 124, 144, 160, 181] [3, 44, 56, 89, 125, 145, 161, 182] [4, 45, 57, 90, 126, 146, 162, 183] [5, 46, 58, 91, 127, 147, 163, 184] [6, 47, 59, 92, 128, 148, 164, 185] [7, 48, 60, 93, 129, 149, 165, 186] [8, 49, 61, 94, 130, 150, 166, 187] [9, 50, 62, 95, 131, 151, 167, 188] [10, 51, 63, 96, 132, 152, 168, 189] [11, 52, 64, 97, 133, 153, 169, 190] [12, 53, 65, 98, 134, 154, 170, 191] [13, 54, 66, 99, 135, 155, 171, 192] [14, 55, 67, 100, 136, 156, 172, 193] [15, 56, 68, 101, 137, 157, 173, 194] [16, 57, 69, 102, 138, 158, 174, 195] [33, 52, 70, 93, 126, 174, 175, 194] [34, 53, 71, 94, 127, 140, 176, 195] [0, 54, 72, 95, 128, 141, 177, 196] [1, 55, 73, 96, 129, 142, 178, 197] [2, 56, 74, 97, 130, 143, 179, 198] [3, 57, 75, 98, 131, 144, 180, 199] [4, 58, 76, 99, 132, 145, 181, 200] [5, 59, 77, 100, 133, 146, 182, 201] [6, 60, 78, 101, 134, 147, 183, 202] [7, 61, 79, 102, 135, 148, 184, 203] [8, 62, 80, 103, 136, 149, 185, 204] [9, 63, 81, 104, 137, 150, 186, 205] [10, 64, 70, 82, 138, 151, 187, 206] [11, 65, 71, 83, 139, 152, 188, 207] [12, 66, 72, 84, 105, 153, 189, 208] [13, 67, 73, 85, 106, 154, 190, 209] [14, 68, 74, 86, 107, 155, 175, 191] [15, 69, 75, 87, 108, 156, 176, 192] [16, 35, 76, 88, 109, 157, 177, 193] [17, 36, 77, 89, 110, 158, 178, 194] [18, 37, 78, 90, 111, 159, 179, 195] [19, 38, 79, 91, 112, 160, 180, 196] [20, 39, 80, 92, 113, 161, 181, 197] [21, 40, 81, 93, 114, 162, 182, 198] [22, 41, 82, 94, 115, 163, 183, 199] [23, 42, 83, 95, 116, 164, 184, 200] [24, 43, 84, 96, 117, 165, 185, 201] [25, 44, 85, 97, 118, 166, 186, 202] [26, 45, 86, 98, 119, 167, 187, 203] [27, 46, 87, 99, 120, 168, 188, 204] [28, 47, 88, 100, 121, 169, 189, 205] [29, 48, 89, 101, 122, 170, 190, 206] [30, 49, 90, 102, 123, 171, 191, 207] [31, 50, 91, 103, 124, 172, 192, 208] [32, 51, 92, 104, 125, 173, 193, 209]

Code ID 210-10-21 · download JSON · raw on GitHub