[[224,18,15]] · QEC Challenge

[[224,18,15]] d ≤

224

kd²/n

18.08

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (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, 15 qubits)

[16, 27, 30, 55, 57, 60, 64, 71, 72, 82, 115, 128, 158, 180, 195]

d_Z 15 · witness weight 15 (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, 15 qubits)

[21, 26, 52, 71, 105, 115, 123, 125, 129, 139, 149, 164, 194, 218, 221]

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 (3,4): 224 (3,5): 1008 (3,6): 11200 (3,7): 17584 (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 (3,4): 224 (3,5): 1008 (3,6): 11200 (3,7): 17584 (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 C28 x C4 = SmallGroup(112, 19) (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, 1, 38, 41}, b = {0, 21, 66} 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

[[224,18,15]] — two-block group-algebra code over C28 x C4

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/224-18-15.json.

Parity checks

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

H_X (112 checks, sparse supports)

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

H_Z (112 checks, sparse supports)

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

Code ID 224-18-15 · download JSON · raw on GitHub