[[200,24,16]] · QEC Challenge

[[200,24,16]] d ≤

200

kd²/n

30.72

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (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, 16 qubits)

[30, 43, 49, 63, 79, 81, 87, 89, 111, 119, 131, 144, 157, 164, 170, 175]

d_Z 16 · witness weight 16 (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, 16 qubits)

[2, 9, 21, 23, 39, 46, 82, 95, 100, 119, 121, 134, 140, 172, 175, 191]

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 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 (2,6)×600 (3,6)×400 (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 (2,6): 600 (2,7): 2000 (2,8): 1000 (3,6): 400 (3,7): 900 (3,8): 6600 (3,9): 32500 (3,10): 40200 (3,11): 15400 (3,12): 4000 (3,13): 1000

trapping sets H_Z (1,4)×100 (2,6)×600 (3,6)×400 (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 (2,6): 600 (2,7): 2000 (2,8): 1000 (3,6): 400 (3,7): 900 (3,8): 6600 (3,9): 32500 (3,10): 40200 (3,11): 15400 (3,12): 4000 (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, 11, 70, 72, 96}, b = {0, 9, 32, 46} 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

[[200,24,16]] — 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/200-24-16.json.

Parity checks

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

H_X (100 checks, sparse supports)

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

H_Z (100 checks, sparse supports)

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

Code ID 200-24-16 · download JSON · raw on GitHub