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[[224,18,12]] d ≤
n
224
k
18
d
12
kd²/n
11.571
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 12 · witness weight 12 (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, 12 qubits)
[19, 33, 35, 50, 51, 65, 66, 67, 81, 82, 96, 108]
d_Z 12 · witness weight 12 (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, 12 qubits)
[112, 116, 125, 126, 141, 143, 172, 173, 188, 191, 206, 217]
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 6 · H_Z 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×224 (2,4)×1680 (3,3)×224 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 224 (2,4): 1680 (3,3): 224 (3,5): 16128 (3,7): 2240
trapping sets H_Z (1,3)×224 (2,4)×1680 (3,3)×224 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 224 (2,4): 1680 (3,3): 224 (3,5): 16128 (3,7): 2240

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C7 x (C4 : C4) = SmallGroup(112, 21) (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, 40}, b = {0, 29, 105} as element indices. Check weight 6. 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 2bga-nonabelian (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[224,18,12]] — two-block group-algebra code over C7 x (C4 : 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/224-18-12.json.

Parity checks

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