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[[224,24,10]] d ≤
n
224
k
24
d
10
kd²/n
10.714
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · 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 10 · witness weight 10 (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, 10 qubits)
[13, 31, 37, 44, 63, 69, 117, 118, 158, 217]
d_Z 10 · witness weight 10 (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, 10 qubits)
[1, 12, 52, 86, 121, 135, 141, 157, 196, 211]
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): 1008 (3,6): 11872 (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 (3,5): 1008 (3,6): 11872 (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 C14 x C2 x C2 x C2 = SmallGroup(112, 43) (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, 19, 44, 78}, b = {0, 21, 60} 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

[[224,24,10]] — two-block group-algebra code over C14 x C2 x C2 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/224-24-10.json.

Parity checks

X-checks 112 (max weight 7) · Z-checks 112 (max weight 7)
H_X (112 checks, sparse supports)
[0, 30, 92, 98, 112, 188, 212] [1, 38, 91, 99, 113, 196, 205] [2, 42, 87, 102, 114, 200, 219] [3, 44, 78, 107, 115, 174, 199] [4, 16, 77, 108, 116, 173, 221] [5, 6, 46, 105, 117, 119, 204] [6, 50, 79, 106, 118, 207, 215] [7, 52, 86, 103, 119, 182, 191] [8, 23, 85, 104, 120, 181, 217] [2, 9, 54, 109, 115, 121, 211] [10, 56, 90, 100, 122, 186, 210] [11, 27, 89, 101, 123, 185, 223] [1, 12, 58, 110, 124, 129, 214] [13, 29, 63, 111, 125, 159, 213] [14, 17, 60, 94, 113, 126, 190] [15, 18, 31, 93, 127, 132, 189] [13, 16, 19, 62, 128, 133, 217] [17, 64, 93, 97, 129, 194, 203] [18, 35, 94, 96, 130, 193, 222] [0, 19, 66, 111, 122, 131, 218] [20, 37, 71, 110, 132, 167, 206] [10, 21, 68, 101, 112, 133, 198] [11, 22, 39, 100, 125, 134, 197] [12, 20, 23, 70, 126, 135, 221] [24, 41, 75, 109, 136, 171, 220] [7, 25, 72, 104, 118, 137, 202] [8, 26, 43, 103, 138, 144, 201] [9, 24, 27, 74, 139, 145, 222] [28, 32, 45, 79, 120, 140, 175] [4, 29, 33, 76, 121, 141, 206] [3, 30, 34, 47, 142, 148, 205] [28, 31, 35, 78, 125, 143, 149] [32, 49, 83, 105, 144, 179, 216] [3, 33, 80, 108, 114, 145, 209] [4, 34, 51, 107, 136, 146, 208] [5, 32, 35, 82, 137, 147, 223] [24, 36, 53, 87, 116, 148, 183] [8, 25, 37, 84, 117, 149, 213] [7, 26, 38, 55, 140, 150, 212] [27, 36, 39, 86, 132, 141, 151] [20, 40, 57, 91, 130, 152, 187] [11, 21, 41, 88, 131, 153, 216] [10, 22, 42, 59, 154, 160, 215] [23, 40, 43, 90, 136, 155, 161] [0, 44, 48, 61, 134, 156, 191] [15, 45, 49, 92, 116, 135, 157] [14, 46, 50, 63, 115, 158, 164] [44, 47, 51, 94, 140, 159, 165] [13, 48, 65, 98, 123, 160, 195] [14, 18, 49, 95, 124, 161, 220] [15, 17, 50, 67, 152, 162, 219] [16, 48, 51, 97, 144, 153, 163] [1, 40, 52, 69, 127, 164, 199] [22, 41, 53, 99, 120, 128, 165] [21, 42, 54, 71, 119, 156, 166] [43, 52, 55, 101, 148, 157, 167] [2, 36, 56, 73, 146, 168, 203] [26, 37, 57, 102, 123, 147, 169] [25, 38, 58, 75, 122, 170, 176] [39, 56, 59, 104, 152, 171, 177] [5, 60, 64, 77, 112, 150, 172] [30, 61, 65, 105, 127, 151, 173] [29, 62, 66, 79, 126, 174, 180] [4, 60, 63, 67, 156, 175, 181] [6, 28, 64, 81, 138, 176, 210] [29, 34, 65, 106, 130, 139, 177] [30, 33, 66, 83, 129, 168, 178] [31, 64, 67, 108, 160, 169, 179] [9, 56, 68, 85, 113, 142, 180] [38, 57, 69, 109, 134, 143, 181] [37, 58, 70, 87, 133, 172, 182] [8, 59, 68, 71, 164, 173, 183] [12, 52, 72, 89, 114, 162, 184] [42, 53, 73, 110, 138, 163, 185] [41, 54, 74, 91, 137, 186, 192] [11, 55, 72, 75, 168, 187, 193] [16, 76, 80, 93, 117, 166, 188] [13, 46, 77, 81, 142, 167, 189] [0, 45, 78, 82, 141, 190, 196] [15, 76, 79, 83, 172, 191, 197] [19, 44, 80, 96, 118, 154, 192] [45, 50, 81, 111, 146, 155, 193] [46, 49, 82, 98, 145, 184, 194] [18, 47, 80, 83, 176, 185, 195] [23, 72, 84, 100, 121, 158, 196] [20, 54, 73, 85, 150, 159, 197] [1, 53, 74, 86, 149, 188, 198] [22, 75, 84, 87, 180, 189, 199] [27, 68, 88, 103, 124, 178, 200] [24, 58, 69, 89, 154, 179, 201] [2, 57, 70, 90, 153, 202, 207] [26, 71, 88, 91, 184, 203, 208] [3, 31, 92, 95, 128, 182, 204] [28, 62, 93, 96, 158, 183, 205] [5, 61, 94, 97, 157, 206, 211] [35, 60, 95, 107, 131, 170, 207] [32, 61, 66, 96, 162, 171, 208] [6, 62, 65, 97, 161, 200, 209] [34, 63, 95, 98, 192, 201, 210] [7, 39, 88, 99, 135, 174, 211] [36, 70, 89, 100, 166, 175, 212] [9, 69, 90, 101, 165, 204, 213] [10, 43, 84, 102, 139, 194, 214] [40, 74, 85, 103, 170, 195, 215] [12, 73, 86, 104, 169, 216, 218] [14, 47, 105, 106, 143, 198, 217] [17, 51, 76, 106, 147, 186, 218] [48, 77, 82, 107, 178, 187, 219] [19, 78, 81, 108, 177, 214, 220] [21, 55, 102, 109, 151, 190, 221] [25, 59, 99, 110, 155, 209, 222] [33, 67, 92, 111, 163, 202, 223]
H_Z (112 checks, sparse supports)
[0, 21, 60, 112, 131, 156, 190] [1, 14, 68, 113, 124, 164, 198] [2, 33, 72, 114, 121, 168, 202] [3, 9, 46, 115, 142, 145, 204] [4, 36, 45, 116, 141, 146, 175] [5, 37, 76, 117, 147, 172, 206] [6, 25, 80, 117, 118, 176, 209] [5, 7, 54, 119, 137, 150, 211] [8, 28, 53, 120, 138, 149, 183] [9, 29, 84, 121, 139, 180, 213] [10, 19, 58, 122, 133, 154, 214] [11, 48, 57, 123, 134, 153, 187] [12, 49, 88, 124, 135, 184, 216] [13, 22, 31, 125, 128, 160, 189] [14, 23, 62, 126, 158, 161, 217] [15, 52, 61, 127, 157, 162, 191] [16, 53, 92, 116, 128, 163, 188] [12, 17, 66, 126, 129, 162, 218] [18, 40, 65, 127, 130, 161, 195] [19, 41, 95, 128, 131, 192, 220] [15, 20, 39, 132, 135, 152, 197] [16, 21, 70, 133, 153, 166, 221] [22, 44, 69, 134, 154, 165, 199] [23, 45, 99, 120, 135, 155, 196] [24, 34, 43, 136, 139, 148, 201] [25, 35, 74, 137, 149, 170, 222] [26, 64, 73, 138, 150, 169, 203] [27, 65, 102, 123, 139, 151, 200] [28, 38, 47, 140, 143, 176, 205] [29, 39, 78, 125, 141, 174, 177] [30, 68, 77, 112, 142, 173, 178] [31, 69, 105, 127, 143, 179, 204] [26, 32, 51, 140, 144, 147, 208] [27, 33, 82, 141, 145, 178, 223] [34, 56, 81, 142, 146, 177, 210] [35, 57, 106, 130, 143, 147, 207] [30, 36, 55, 148, 151, 168, 212] [31, 37, 86, 132, 149, 169, 182] [38, 60, 85, 113, 150, 170, 181] [39, 61, 109, 134, 151, 171, 211] [40, 50, 59, 152, 155, 164, 215] [41, 51, 90, 136, 153, 165, 186] [42, 80, 89, 114, 154, 166, 185] [43, 81, 110, 138, 155, 167, 214] [44, 54, 63, 115, 156, 159, 192] [45, 55, 94, 140, 157, 190, 193] [46, 84, 93, 117, 158, 189, 194] [13, 47, 85, 142, 159, 195, 217] [42, 48, 67, 156, 160, 163, 219] [43, 49, 97, 144, 157, 161, 194] [50, 72, 96, 118, 158, 162, 193] [51, 73, 111, 146, 159, 163, 218] [46, 52, 71, 119, 164, 167, 184] [47, 53, 101, 148, 165, 185, 198] [54, 76, 100, 121, 166, 186, 197] [20, 55, 77, 150, 167, 187, 221] [56, 66, 75, 122, 168, 171, 180] [57, 67, 104, 152, 169, 181, 202] [58, 95, 103, 124, 170, 182, 201] [24, 59, 96, 154, 171, 183, 222] [60, 70, 79, 126, 172, 175, 207] [4, 61, 71, 156, 173, 206, 208] [3, 62, 99, 128, 174, 205, 209] [28, 63, 100, 125, 158, 175, 210] [58, 64, 83, 129, 172, 176, 179] [59, 65, 108, 160, 173, 177, 209] [66, 88, 107, 131, 174, 178, 208] [32, 67, 89, 162, 175, 179, 223] [62, 68, 87, 133, 180, 183, 200] [8, 63, 69, 164, 181, 201, 213] [7, 70, 92, 135, 182, 202, 212] [36, 71, 93, 132, 166, 183, 203] [72, 82, 91, 137, 184, 187, 196] [11, 73, 83, 168, 185, 197, 216] [10, 74, 106, 139, 186, 198, 215] [40, 75, 107, 136, 170, 187, 199] [0, 76, 86, 141, 188, 191, 218] [15, 77, 87, 116, 172, 189, 219] [14, 78, 109, 115, 143, 190, 220] [7, 44, 79, 118, 140, 174, 191] [74, 80, 98, 145, 188, 192, 195] [18, 75, 81, 176, 189, 193, 220] [17, 82, 102, 147, 190, 194, 219] [48, 83, 103, 144, 178, 191, 195] [1, 78, 84, 149, 196, 199, 214] [22, 79, 85, 120, 180, 197, 215] [21, 86, 105, 119, 151, 198, 216] [3, 52, 87, 114, 148, 182, 199] [2, 88, 97, 153, 200, 203, 211] [26, 89, 98, 123, 184, 201, 212] [25, 90, 111, 122, 155, 202, 213] [17, 56, 91, 113, 152, 186, 203] [5, 92, 101, 112, 157, 204, 223] [1, 30, 93, 127, 129, 188, 205] [20, 29, 94, 126, 130, 159, 206] [6, 90, 95, 161, 204, 207, 210] [34, 91, 96, 130, 192, 205, 208] [33, 97, 110, 129, 163, 206, 209] [10, 64, 98, 112, 160, 194, 210] [9, 94, 99, 113, 165, 211, 222] [0, 38, 100, 122, 134, 196, 212] [13, 37, 101, 123, 133, 167, 213] [12, 102, 108, 114, 169, 214, 221] [6, 42, 103, 119, 138, 200, 215] [32, 41, 104, 120, 137, 171, 216] [8, 16, 105, 117, 144, 173, 217] [19, 104, 106, 118, 177, 217, 218] [2, 50, 107, 115, 146, 207, 219] [24, 49, 108, 116, 145, 179, 220] [4, 23, 109, 121, 136, 181, 221] [18, 27, 110, 124, 132, 185, 222] [11, 35, 111, 125, 131, 193, 223]
Code ID 224-24-10 · download JSON · raw on GitHub