[[224,6,21]] · QEC Challenge

[[224,6,21]] d ≤

224

kd²/n

11.812

X/Z

Share this result

Distance

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

[3, 10, 28, 48, 51, 54, 64, 71, 89, 95, 105, 110, 135, 147, 152, 156, 176, 180, 197, 213, 217]

d_Z 21 · witness weight 21 (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, 21 qubits)

[12, 19, 32, 33, 46, 51, 61, 79, 101, 103, 111, 122, 128, 136, 146, 176, 177, 197, 213, 215, 218]

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 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, 7, 13, 108}, b = {0, 10, 48} 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

[[224,6,21]] — 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/224-6-21.json.

Parity checks

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

H_X (112 checks, sparse supports)

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

H_Z (112 checks, sparse supports)

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

Code ID 224-6-21 · download JSON · raw on GitHub