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[[228,4,17]] d ≤
n
228
k
4
d
17
kd²/n
5.07
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 17 · witness weight 17 (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, 17 qubits)
[1, 19, 27, 40, 58, 62, 69, 82, 91, 99, 110, 133, 134, 174, 185, 187, 196]
d_Z 17 · witness weight 17 (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, 17 qubits)
[79, 88, 90, 99, 100, 101, 121, 122, 134, 142, 163, 164, 176, 184, 193, 206, 214]
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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×114 (2,2)×114 (3,2)×114 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 114 (1,3): 114 (2,2): 114 (2,3): 684 (2,4): 342 (3,2): 114 (3,3): 2052 (3,4): 3762 (3,5): 1710 (3,6): 684 (3,7): 114
trapping sets H_Z (1,2)×114 (2,2)×114 (3,2)×114 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 114 (1,3): 114 (2,2): 114 (2,3): 684 (2,4): 342 (3,2): 114 (3,3): 2052 (3,4): 3762 (3,5): 1710 (3,6): 684 (3,7): 114

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C19 x S3 = SmallGroup(114, 3) (order 114), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 228; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 4, 16}, b = {0, 36} as element indices. Check weight 5. 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

[[228,4,17]] — two-block group-algebra code over C19 x S3

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| = 228

Parameters are also spelled out in provenance.construction of codes/228-4-17.json.

Parity checks

X-checks 114 (max weight 5) · Z-checks 114 (max weight 5)
H_X (114 checks, sparse supports)
[0, 105, 106, 114, 180] [1, 102, 109, 115, 183] [1, 2, 110, 116, 186] [3, 110, 111, 117, 187] [0, 4, 112, 118, 189] [5, 96, 108, 119, 190] [4, 5, 6, 120, 192] [5, 7, 113, 121, 193] [8, 99, 113, 122, 194] [2, 8, 9, 123, 195] [3, 10, 102, 124, 196] [11, 103, 112, 125, 197] [9, 10, 12, 126, 198] [10, 11, 13, 127, 199] [11, 14, 105, 128, 200] [6, 14, 15, 129, 201] [0, 7, 16, 130, 202] [8, 17, 108, 131, 203] [15, 16, 18, 132, 204] [16, 17, 19, 133, 205] [1, 17, 20, 134, 206] [12, 20, 21, 135, 207] [2, 13, 22, 136, 208] [3, 14, 23, 137, 209] [21, 22, 24, 138, 210] [22, 23, 25, 139, 211] [4, 23, 26, 140, 212] [18, 26, 27, 141, 213] [6, 19, 28, 142, 214] [7, 20, 29, 143, 215] [27, 28, 30, 144, 216] [28, 29, 31, 145, 217] [9, 29, 32, 146, 218] [24, 32, 33, 147, 219] [12, 25, 34, 148, 220] [13, 26, 35, 149, 221] [33, 34, 36, 114, 150] [34, 35, 37, 151, 222] [15, 35, 38, 152, 223] [30, 38, 39, 115, 153] [18, 31, 40, 154, 224] [19, 32, 41, 155, 225] [39, 40, 42, 116, 156] [40, 41, 43, 117, 157] [21, 41, 44, 158, 226] [36, 44, 45, 118, 159] [24, 37, 46, 119, 160] [25, 38, 47, 161, 227] [45, 46, 48, 120, 162] [46, 47, 49, 121, 163] [27, 47, 50, 122, 164] [42, 50, 51, 123, 165] [30, 43, 52, 124, 166] [31, 44, 53, 125, 167] [51, 52, 54, 126, 168] [52, 53, 55, 127, 169] [33, 53, 56, 128, 170] [48, 56, 57, 129, 171] [36, 49, 58, 130, 172] [37, 50, 59, 131, 173] [57, 58, 60, 132, 174] [58, 59, 61, 133, 175] [39, 59, 62, 134, 176] [54, 62, 63, 135, 177] [42, 55, 64, 136, 178] [43, 56, 65, 137, 179] [63, 64, 66, 138, 180] [64, 65, 67, 139, 181] [45, 65, 68, 140, 182] [60, 68, 69, 141, 183] [48, 61, 70, 142, 184] [49, 62, 71, 143, 185] [69, 70, 72, 144, 186] [70, 71, 73, 145, 187] [51, 71, 74, 146, 188] [66, 74, 75, 147, 189] [54, 67, 76, 148, 190] [55, 68, 77, 149, 191] [75, 76, 78, 150, 192] [76, 77, 79, 151, 193] [57, 77, 80, 152, 194] [72, 80, 81, 153, 195] [60, 73, 82, 154, 196] [61, 74, 83, 155, 197] [81, 82, 84, 156, 198] [82, 83, 85, 157, 199] [63, 83, 86, 158, 200] [78, 86, 87, 159, 201] [66, 79, 88, 160, 202] [67, 80, 89, 161, 203] [87, 88, 90, 162, 204] [88, 89, 91, 163, 205] [69, 89, 92, 164, 206] [84, 92, 93, 165, 207] [72, 85, 94, 166, 208] [73, 86, 95, 167, 209] [93, 94, 96, 168, 210] [94, 95, 97, 169, 211] [75, 95, 98, 170, 212] [90, 98, 99, 171, 213] [78, 91, 100, 172, 214] [79, 92, 101, 173, 215] [99, 100, 102, 174, 216] [100, 101, 103, 175, 217] [81, 101, 104, 176, 218] [96, 104, 105, 177, 219] [84, 97, 106, 178, 220] [85, 98, 107, 179, 221] [106, 107, 108, 181, 222] [87, 107, 109, 182, 223] [90, 103, 110, 184, 224] [91, 104, 111, 185, 225] [93, 111, 112, 188, 226] [97, 109, 113, 191, 227]
H_Z (114 checks, sparse supports)
[0, 36, 114, 118, 130] [1, 39, 115, 116, 134] [2, 42, 116, 123, 136] [3, 43, 117, 124, 137] [4, 45, 118, 120, 140] [5, 46, 119, 120, 121] [6, 48, 120, 129, 142] [7, 49, 121, 130, 143] [8, 50, 122, 123, 131] [9, 51, 123, 126, 146] [10, 52, 124, 126, 127] [11, 53, 125, 127, 128] [12, 54, 126, 135, 148] [13, 55, 127, 136, 149] [14, 56, 128, 129, 137] [15, 57, 129, 132, 152] [16, 58, 130, 132, 133] [17, 59, 131, 133, 134] [18, 60, 132, 141, 154] [19, 61, 133, 142, 155] [20, 62, 134, 135, 143] [21, 63, 135, 138, 158] [22, 64, 136, 138, 139] [23, 65, 137, 139, 140] [24, 66, 138, 147, 160] [25, 67, 139, 148, 161] [26, 68, 140, 141, 149] [27, 69, 141, 144, 164] [28, 70, 142, 144, 145] [29, 71, 143, 145, 146] [30, 72, 144, 153, 166] [31, 73, 145, 154, 167] [32, 74, 146, 147, 155] [33, 75, 147, 150, 170] [34, 76, 148, 150, 151] [35, 77, 149, 151, 152] [36, 78, 150, 159, 172] [37, 79, 151, 160, 173] [38, 80, 152, 153, 161] [39, 81, 153, 156, 176] [40, 82, 154, 156, 157] [41, 83, 155, 157, 158] [42, 84, 156, 165, 178] [43, 85, 157, 166, 179] [44, 86, 158, 159, 167] [45, 87, 159, 162, 182] [46, 88, 160, 162, 163] [47, 89, 161, 163, 164] [48, 90, 162, 171, 184] [49, 91, 163, 172, 185] [50, 92, 164, 165, 173] [51, 93, 165, 168, 188] [52, 94, 166, 168, 169] [53, 95, 167, 169, 170] [54, 96, 168, 177, 190] [55, 97, 169, 178, 191] [56, 98, 170, 171, 179] [57, 99, 171, 174, 194] [58, 100, 172, 174, 175] [59, 101, 173, 175, 176] [60, 102, 174, 183, 196] [61, 103, 175, 184, 197] [62, 104, 176, 177, 185] [63, 105, 177, 180, 200] [64, 106, 178, 180, 181] [65, 107, 179, 181, 182] [0, 66, 180, 189, 202] [67, 108, 181, 190, 203] [68, 109, 182, 183, 191] [1, 69, 183, 186, 206] [70, 110, 184, 186, 187] [71, 111, 185, 187, 188] [2, 72, 186, 195, 208] [3, 73, 187, 196, 209] [74, 112, 188, 189, 197] [4, 75, 189, 192, 212] [5, 76, 190, 192, 193] [77, 113, 191, 193, 194] [6, 78, 192, 201, 214] [7, 79, 193, 202, 215] [8, 80, 194, 195, 203] [9, 81, 195, 198, 218] [10, 82, 196, 198, 199] [11, 83, 197, 199, 200] [12, 84, 198, 207, 220] [13, 85, 199, 208, 221] [14, 86, 200, 201, 209] [15, 87, 201, 204, 223] [16, 88, 202, 204, 205] [17, 89, 203, 205, 206] [18, 90, 204, 213, 224] [19, 91, 205, 214, 225] [20, 92, 206, 207, 215] [21, 93, 207, 210, 226] [22, 94, 208, 210, 211] [23, 95, 209, 211, 212] [24, 96, 119, 210, 219] [25, 97, 211, 220, 227] [26, 98, 212, 213, 221] [27, 99, 122, 213, 216] [28, 100, 214, 216, 217] [29, 101, 215, 217, 218] [30, 102, 115, 124, 216] [31, 103, 125, 217, 224] [32, 104, 218, 219, 225] [33, 105, 114, 128, 219] [34, 106, 114, 220, 222] [35, 107, 221, 222, 223] [37, 108, 119, 131, 222] [38, 109, 115, 223, 227] [40, 110, 116, 117, 224] [41, 111, 117, 225, 226] [44, 112, 118, 125, 226] [47, 113, 121, 122, 227]
Code ID 228-4-17 · download JSON · raw on GitHub