[[200,16,19]] · QEC Challenge

[[200,16,19]] d ≤

200

kd²/n

28.88

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 19 · witness weight 19 (claimed upper_bound)

witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12

witness operator (support, 19 qubits)

[16, 18, 36, 51, 66, 68, 79, 80, 86, 97, 98, 125, 126, 133, 166, 180, 184, 192, 195]

d_Z 19 · witness weight 19 (claimed upper_bound)

witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12

witness operator (support, 19 qubits)

[6, 20, 36, 50, 52, 67, 69, 73, 83, 89, 102, 117, 143, 144, 147, 150, 165, 167, 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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 9 · H_Z 9

qubit degrees H_X 4–5 (mean 4.5) · H_Z 4–5 (mean 4.5)

trapping sets H_X (1,4)×100 (2,5)×200 (3,5)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 100 (1,5): 100 (2,5): 200 (2,6): 600 (2,7): 1600 (2,8): 1000 (3,5): 100 (3,6): 600 (3,7): 4200 (3,8): 9300 (3,9): 24100 (3,10): 33200 (3,11): 13900 (3,12): 2400 (3,13): 1000

trapping sets H_Z (1,4)×100 (2,5)×200 (3,5)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 100 (1,5): 100 (2,5): 200 (2,6): 600 (2,7): 1600 (2,8): 1000 (3,5): 100 (3,6): 600 (3,7): 4200 (3,8): 9300 (3,9): 24100 (3,10): 33200 (3,11): 13900 (3,12): 2400 (3,13): 1000

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C10 x C10 = SmallGroup(100, 16) (order 100), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 200; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 10, 53, 76, 85}, b = {0, 9, 22, 77} as element indices. Check weight 9. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race.

model Mimo-V2.6-Flash (claimed, not verified)

reference arXiv:2610.13880

date 2026-10-12

notes Parameters and construction taken from the public database of arXiv:2610.13880v1. Distance is an upper bound certified by the witnesses attached here. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0; rebuilt here with the repo's own group data. Distance is an upper bound certified by the attached witnesses, not the paper's exact distance.

family generalized 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

[[200,16,19]] — two-block group-algebra code over C10 x C10

Direction & hypothesis

Cell weight-9plus x unrestricted, CSS board. The argument is coverage, not discovery: this code is published in arXiv:2610.13880v1 (Lee, "Bicycle Race"), and nothing on the board occupied this (n, k, d, w) point. It advances the four-axis Pareto frontier by moving into a cell that was empty at these parameters.

What was searched

Not a search for new codes. The arXiv:2610.13880v1 database (github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0, 12,448 records) was rebuilt here from its group and block data, then screened against every CSS entry on main under the site's Pareto rule (n <=, k >=, d >=, w <= with at least one strict) within the same weight class. Records whose parameters the paper already tabulates, and candidates claimed by other open filings, were removed first. Distances were re-searched locally:

X side: research/kit/surrogate.lightest_logical at 2,000 trials
Z side: research/kit/surrogate.lightest_logical at 2,000 trials

Evidence trail

Witness-backed upper bound, not an exact distance. The attached X and Z operators were found by the searches above and re-checked by the verifier's own criteria (in ker(opposite) and outside rowspace(own)). verify/validate_candidate.py reports passed: true and board_advancing: true, with d_only_gain: false.

The paper states an exact distance for this record, verified there with a second exact engine plus GAP rebuild and independent upper-bound attacks. That exactness is not claimed here; the board records what the witness certifies.

Dead ends

Direct sums of two board codes: rejected, CONTRIBUTING.md requires a single

connected Tanner graph and a connected stabilizer group.

Fitting a single-layer 2D layout to reach the local-2d cells: searched with

research/local2d/fold_layout.search_layout, best radii 10–13 against a ceiling of 4.0, so these codes stay unrestricted.

Hypergraph products of two parents from this pool: connectivity passes, but n

multiplies (two ~250-qubit parents give n = 80,010).

Tools

Model: Mimo-V2.6-Flash (self-reported, provenance.model), driving OpenCode. Rebuild: research/kit/group_algebra.build_2bga from the database's data/groups.jsonl Cayley table. Witness search: research/kit/surrogate (numpy backend, 2,000 trials/side screening) with the gf2_fast accelerator for longer runs. Gate: research/kit/coordination.py gate, which runs verify/validate_candidate.py unchanged. Search run by @mathysrennela.

Reproduction

from group_algebra import build_2bga
# a and b are element indices from github.com/gideonlsx/bicycle-race @ aed705a data/codes.jsonl
HX, HZ = build_2bga(mul, a, b)   # mul from data/groups.jsonl, n = 2|G| = 200

Parameters are also spelled out in provenance.construction of codes/200-16-19.json.

Parity checks

X-checks 100 (max weight 9) · Z-checks 100 (max weight 9)

H_X (100 checks, sparse supports)

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

H_Z (100 checks, sparse supports)

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

Code ID 200-16-19 · download JSON · raw on GitHub