[[224,12,19]] · QEC Challenge

[[224,12,19]] d ≤

224

kd²/n

19.339

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · w_X = 7, w_Z = 7 (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)

[2, 7, 55, 63, 66, 70, 73, 78, 88, 97, 119, 125, 133, 149, 175, 182, 194, 204, 217]

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)

[28, 35, 36, 52, 97, 102, 106, 115, 138, 142, 145, 153, 176, 187, 188, 196, 197, 216, 220]

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 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×112 (2,4)×336 (3,3)×112 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 112 (1,4): 112 (2,4): 336 (2,5): 1344 (2,6): 672 (3,3): 112 (3,5): 1232 (3,6): 11872 (3,7): 16912 (3,8): 6048 (3,9): 2016 (3,10): 448

trapping sets H_Z (1,3)×112 (2,4)×336 (3,3)×112 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 112 (1,4): 112 (2,4): 336 (2,5): 1344 (2,6): 672 (3,3): 112 (3,5): 1232 (3,6): 11872 (3,7): 16912 (3,8): 6048 (3,9): 2016 (3,10): 448

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C56 x C2 = SmallGroup(112, 22) (order 112), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 224; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 3, 91, 99}, b = {0, 25, 54} as element indices. Check weight 7. 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

[[224,12,19]] — two-block group-algebra code over C56 x C2

Direction & hypothesis

Cell weight-8 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| = 224

Parameters are also spelled out in provenance.construction of codes/224-12-19.json.

Parity checks

X-checks 112 (max weight 7) · Z-checks 112 (max weight 7)

H_X (112 checks, sparse supports)

[0, 22, 57, 76, 112, 181, 222] [1, 31, 65, 84, 113, 174, 223] [2, 34, 45, 88, 114, 193, 217] [0, 3, 38, 73, 115, 139, 197] [4, 39, 58, 92, 116, 182, 200] [5, 7, 75, 93, 117, 199, 214] [6, 43, 53, 95, 118, 186, 221] [1, 7, 47, 81, 119, 147, 190] [3, 8, 66, 99, 120, 191, 207] [9, 14, 83, 100, 121, 156, 218] [2, 10, 50, 61, 122, 128, 208] [11, 46, 51, 102, 123, 188, 194] [12, 17, 63, 103, 124, 204, 210] [3, 13, 54, 89, 125, 155, 211] [4, 14, 55, 74, 114, 126, 198] [5, 15, 20, 91, 123, 127, 213] [16, 21, 40, 105, 128, 164, 215] [6, 17, 59, 69, 129, 135, 202] [10, 18, 54, 106, 130, 196, 203] [19, 25, 71, 107, 131, 168, 211] [7, 20, 63, 96, 132, 163, 205] [8, 13, 21, 82, 118, 133, 206] [9, 22, 29, 98, 130, 134, 172] [15, 23, 48, 109, 135, 173, 219] [10, 24, 66, 77, 136, 143, 218] [11, 25, 62, 67, 112, 137, 209] [12, 26, 32, 79, 116, 138, 220] [27, 28, 33, 110, 139, 176, 205] [13, 28, 70, 102, 120, 140, 171] [14, 29, 71, 90, 122, 141, 212] [15, 30, 36, 104, 137, 142, 221] [16, 31, 37, 56, 124, 143, 180] [17, 32, 75, 85, 144, 151, 215] [18, 24, 33, 70, 113, 145, 216] [19, 34, 41, 87, 120, 146, 184] [26, 35, 36, 111, 147, 185, 212] [20, 36, 79, 106, 117, 148, 179] [21, 28, 37, 97, 129, 149, 217] [22, 38, 45, 108, 145, 150, 188] [23, 30, 39, 64, 131, 151, 189] [24, 40, 82, 92, 130, 152, 159] [25, 41, 78, 83, 115, 153, 219] [26, 42, 48, 94, 126, 154, 223] [27, 43, 44, 49, 117, 155, 192] [11, 28, 44, 86, 133, 156, 187] [29, 45, 87, 103, 121, 136, 157] [30, 46, 52, 110, 135, 153, 158] [31, 47, 53, 72, 138, 159, 196] [32, 48, 91, 99, 124, 160, 167] [33, 40, 49, 86, 119, 161, 222] [34, 50, 57, 101, 133, 162, 200] [35, 42, 51, 52, 121, 163, 201] [18, 36, 52, 94, 127, 164, 195] [37, 44, 53, 107, 128, 144, 165] [38, 54, 61, 111, 112, 161, 166] [39, 46, 55, 80, 146, 167, 204] [4, 40, 56, 97, 145, 168, 175] [41, 57, 93, 98, 125, 131, 169] [42, 58, 64, 105, 141, 147, 170] [43, 59, 60, 65, 127, 171, 207] [25, 44, 60, 100, 149, 172, 203] [12, 45, 61, 101, 134, 152, 173] [27, 46, 62, 68, 151, 169, 174] [47, 63, 69, 88, 113, 154, 175] [8, 48, 64, 104, 138, 176, 183] [49, 56, 65, 100, 132, 139, 177] [50, 66, 73, 109, 114, 149, 178] [51, 58, 67, 68, 134, 179, 214] [33, 52, 68, 105, 142, 180, 210] [19, 53, 60, 69, 143, 160, 181] [35, 54, 70, 77, 115, 177, 182] [55, 62, 71, 95, 116, 162, 183] [14, 56, 72, 107, 161, 184, 191] [5, 57, 73, 108, 140, 146, 185] [16, 58, 74, 80, 157, 163, 186] [59, 75, 76, 81, 118, 142, 187] [9, 41, 60, 76, 165, 188, 216] [26, 61, 77, 109, 150, 168, 189] [43, 62, 78, 84, 167, 185, 190] [2, 63, 79, 85, 119, 170, 191] [21, 64, 80, 110, 154, 192, 199] [9, 65, 72, 81, 148, 155, 193] [23, 66, 82, 89, 122, 165, 194] [67, 74, 83, 84, 123, 150, 195] [16, 49, 68, 84, 158, 196, 220] [34, 69, 76, 85, 159, 176, 197] [51, 70, 86, 92, 125, 193, 198] [6, 71, 78, 87, 126, 178, 199] [19, 29, 72, 88, 177, 200, 206] [15, 73, 89, 111, 156, 162, 201] [31, 74, 90, 95, 173, 179, 202] [0, 75, 91, 96, 129, 158, 203] [23, 42, 77, 92, 166, 184, 204] [1, 59, 78, 93, 183, 201, 205] [10, 79, 94, 99, 132, 186, 206] [27, 37, 80, 95, 170, 207, 213] [22, 81, 88, 96, 164, 171, 208] [39, 82, 97, 102, 136, 181, 209] [1, 83, 90, 98, 137, 166, 210] [0, 50, 85, 99, 175, 192, 211] [4, 67, 86, 100, 140, 208, 212] [17, 87, 93, 101, 141, 194, 213] [30, 35, 89, 102, 172, 178, 214] [6, 47, 90, 103, 189, 195, 215] [3, 91, 104, 106, 144, 174, 216] [8, 24, 94, 105, 148, 202, 217] [2, 38, 96, 106, 180, 187, 218] [11, 55, 97, 107, 152, 197, 219] [7, 98, 103, 108, 153, 182, 220] [5, 32, 101, 109, 157, 209, 221] [13, 18, 104, 110, 160, 190, 222] [12, 20, 108, 111, 169, 198, 223]

H_Z (112 checks, sparse supports)

[0, 25, 54, 112, 115, 203, 211] [1, 33, 63, 113, 119, 205, 210] [2, 14, 66, 114, 122, 191, 218] [3, 41, 70, 115, 120, 125, 216] [4, 26, 71, 116, 126, 168, 212] [5, 36, 43, 117, 127, 185, 221] [6, 21, 75, 118, 129, 199, 215] [7, 49, 79, 117, 119, 132, 220] [8, 28, 34, 120, 133, 176, 217] [9, 45, 51, 121, 134, 188, 193] [10, 29, 82, 122, 130, 136, 206] [11, 15, 83, 123, 137, 156, 219] [12, 31, 48, 124, 138, 173, 223] [13, 57, 86, 125, 133, 140, 222] [14, 42, 87, 121, 126, 141, 184] [15, 52, 59, 127, 135, 142, 201] [10, 16, 53, 128, 143, 186, 196] [17, 37, 91, 124, 129, 144, 213] [18, 22, 40, 130, 145, 164, 222] [19, 39, 57, 131, 146, 181, 200] [20, 65, 94, 127, 132, 148, 223] [21, 44, 50, 128, 133, 149, 192] [22, 61, 67, 112, 134, 150, 208] [17, 23, 46, 135, 151, 194, 204] [24, 45, 97, 136, 145, 152, 217] [25, 30, 98, 131, 137, 153, 172] [26, 47, 64, 138, 147, 154, 189] [3, 27, 65, 139, 155, 174, 207] [28, 73, 100, 139, 140, 149, 156] [29, 58, 101, 134, 141, 157, 200] [30, 68, 75, 142, 151, 158, 214] [24, 31, 69, 113, 143, 159, 202] [32, 53, 104, 138, 144, 160, 221] [33, 38, 56, 139, 145, 161, 180] [34, 55, 73, 114, 146, 162, 197] [7, 35, 58, 147, 163, 182, 214] [36, 81, 105, 142, 147, 148, 164] [37, 60, 66, 143, 149, 165, 207] [38, 77, 83, 115, 150, 166, 218] [32, 39, 62, 116, 151, 167, 209] [40, 61, 107, 128, 152, 161, 168] [41, 46, 108, 146, 153, 169, 188] [42, 63, 80, 154, 163, 170, 204] [13, 43, 81, 118, 155, 171, 190] [9, 44, 89, 155, 156, 165, 172] [45, 74, 109, 114, 150, 157, 173] [46, 84, 91, 123, 158, 167, 174] [40, 47, 85, 119, 159, 175, 215] [48, 69, 110, 135, 154, 160, 176] [49, 54, 72, 155, 161, 177, 196] [50, 71, 89, 122, 162, 178, 211] [20, 51, 74, 123, 163, 179, 198] [16, 52, 96, 158, 163, 164, 180] [53, 76, 82, 118, 159, 165, 181] [54, 92, 98, 125, 130, 166, 182] [48, 55, 78, 126, 167, 183, 219] [19, 56, 77, 143, 168, 177, 184] [57, 62, 111, 112, 162, 169, 185] [58, 79, 95, 116, 170, 179, 186] [28, 59, 96, 129, 171, 187, 205] [22, 60, 102, 171, 172, 181, 188] [23, 61, 90, 122, 166, 173, 189] [1, 62, 104, 137, 174, 183, 190] [56, 63, 99, 124, 132, 175, 191] [27, 64, 85, 151, 170, 176, 192] [65, 70, 88, 113, 171, 177, 193] [66, 87, 102, 120, 136, 178, 194] [36, 67, 90, 137, 179, 195, 212] [31, 68, 106, 174, 179, 180, 196] [0, 69, 97, 129, 175, 181, 197] [4, 70, 108, 140, 145, 182, 198] [64, 71, 93, 131, 141, 183, 199] [34, 72, 92, 159, 184, 193, 200] [35, 73, 78, 115, 178, 185, 201] [6, 74, 94, 126, 186, 195, 202] [44, 75, 106, 117, 144, 187, 203] [11, 38, 76, 112, 187, 188, 197] [39, 77, 103, 136, 182, 189, 204] [7, 78, 110, 153, 190, 199, 205] [8, 72, 79, 138, 148, 191, 206] [43, 80, 99, 167, 186, 192, 207] [2, 81, 86, 119, 187, 193, 208] [11, 82, 101, 133, 152, 194, 209] [52, 83, 103, 121, 153, 195, 210] [18, 47, 84, 113, 190, 195, 196] [3, 85, 107, 144, 191, 197, 211] [14, 86, 111, 156, 161, 198, 212] [5, 80, 87, 146, 157, 199, 213] [4, 50, 88, 114, 175, 200, 208] [51, 89, 93, 125, 194, 201, 214] [17, 90, 105, 141, 202, 210, 215] [18, 60, 91, 127, 160, 203, 216] [12, 55, 92, 116, 152, 198, 204] [20, 27, 93, 117, 169, 205, 213] [21, 88, 94, 154, 164, 206, 217] [8, 59, 95, 118, 183, 202, 207] [10, 96, 100, 132, 203, 208, 218] [25, 97, 109, 149, 168, 209, 219] [12, 68, 98, 134, 169, 210, 220] [13, 19, 99, 120, 160, 206, 211] [29, 35, 100, 121, 172, 177, 212] [15, 95, 101, 162, 173, 213, 221] [5, 67, 102, 123, 140, 209, 214] [16, 32, 103, 124, 157, 215, 220] [33, 76, 104, 142, 176, 216, 222] [2, 37, 105, 128, 170, 180, 217] [9, 24, 106, 130, 148, 216, 218] [23, 41, 107, 131, 165, 184, 219] [26, 84, 108, 150, 185, 220, 223] [6, 30, 109, 135, 178, 189, 221] [0, 49, 110, 139, 158, 192, 222] [1, 42, 111, 147, 166, 201, 223]

Code ID 224-12-19 · download JSON · raw on GitHub