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[[252,22,19]] d ≤
n
252
k
22
d
19
kd²/n
31.516
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 19 · witness weight 19 (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, 19 qubits)
[5, 7, 17, 28, 31, 33, 45, 83, 99, 102, 126, 134, 154, 215, 224, 226, 233, 235, 249]
d_Z 19 · witness weight 19 (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, 19 qubits)
[12, 24, 35, 58, 74, 87, 114, 121, 133, 141, 159, 171, 176, 177, 192, 196, 209, 247, 251]
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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×252 (2,6)×3528 (3,6)×1890 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 252 (2,6): 3528 (3,6): 1890 (3,8): 68418 (3,10): 7056
trapping sets H_Z (1,4)×252 (2,6)×3528 (3,6)×1890 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 252 (2,6): 3528 (3,6): 1890 (3,8): 68418 (3,10): 7056

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C126 = SmallGroup(126, 6) (order 126), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 252; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 28, 68, 79}, b = {0, 9, 80, 106} as element indices. Check weight 8. 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

[[252,22,19]] — two-block group-algebra code over C126

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| = 252

Parameters are also spelled out in provenance.construction of codes/252-22-19.json.

Parity checks

X-checks 126 (max weight 8) · Z-checks 126 (max weight 8)
H_X (126 checks, sparse supports)
[0, 44, 98, 123, 126, 167, 169, 250] [1, 53, 90, 120, 127, 176, 178, 251] [2, 59, 88, 125, 128, 182, 184, 204] [3, 32, 62, 112, 129, 164, 185, 187] [4, 11, 101, 113, 130, 135, 188, 232] [5, 68, 79, 124, 131, 191, 193, 213] [6, 24, 71, 106, 132, 173, 194, 196] [7, 17, 93, 107, 133, 141, 197, 238] [8, 74, 87, 102, 129, 134, 199, 219] [9, 47, 77, 104, 126, 135, 200, 202] [10, 22, 105, 112, 136, 146, 203, 222] [11, 14, 50, 80, 137, 182, 203, 205] [5, 12, 25, 121, 134, 138, 148, 206] [13, 26, 115, 122, 139, 149, 151, 244] [14, 78, 83, 94, 132, 140, 208, 227] [15, 38, 86, 96, 127, 141, 209, 211] [16, 30, 97, 106, 142, 154, 212, 230] [8, 17, 41, 89, 143, 191, 212, 214] [2, 18, 33, 118, 140, 144, 156, 215] [19, 34, 109, 119, 145, 157, 159, 247] [1, 20, 92, 115, 128, 137, 146, 217] [21, 36, 104, 116, 138, 147, 218, 235] [7, 22, 65, 95, 129, 148, 218, 220] [14, 23, 39, 117, 130, 149, 162, 221] [24, 40, 69, 121, 150, 163, 165, 237] [25, 28, 68, 97, 151, 200, 221, 222] [15, 26, 29, 42, 146, 152, 165, 223] [16, 27, 43, 125, 147, 153, 166, 168] [0, 28, 100, 109, 131, 143, 154, 225] [29, 45, 96, 110, 144, 155, 226, 241] [4, 30, 56, 103, 132, 156, 226, 228] [8, 31, 48, 111, 133, 157, 171, 229] [32, 49, 60, 118, 158, 172, 174, 243] [20, 33, 59, 105, 159, 209, 229, 230] [9, 21, 34, 51, 154, 160, 174, 231] [10, 35, 52, 124, 155, 161, 175, 177] [6, 16, 36, 108, 135, 151, 162, 233] [7, 37, 54, 123, 136, 152, 163, 234] [38, 55, 84, 117, 153, 164, 180, 246] [18, 39, 83, 110, 137, 165, 234, 235] [19, 28, 40, 57, 138, 166, 180, 236] [29, 41, 58, 87, 139, 167, 181, 183] [42, 45, 86, 111, 130, 168, 218, 236] [30, 43, 46, 60, 162, 169, 183, 237] [31, 44, 47, 61, 163, 170, 184, 186] [3, 10, 45, 114, 141, 159, 171, 239] [4, 46, 63, 120, 142, 160, 172, 240] [47, 64, 75, 111, 161, 173, 189, 249] [12, 48, 74, 116, 143, 174, 240, 241] [13, 20, 49, 66, 144, 175, 189, 242] [21, 50, 67, 78, 145, 176, 190, 192] [36, 51, 77, 117, 133, 177, 226, 242] [22, 37, 52, 69, 171, 178, 192, 243] [23, 38, 53, 70, 172, 179, 193, 195] [17, 31, 54, 119, 148, 168, 180, 244] [18, 32, 55, 72, 149, 169, 181, 245] [19, 56, 73, 101, 150, 170, 182, 198] [34, 57, 100, 120, 136, 151, 183, 245] [35, 45, 58, 75, 152, 184, 198, 246] [1, 46, 59, 76, 153, 185, 199, 201] [13, 60, 63, 103, 138, 186, 234, 246] [48, 61, 64, 78, 139, 180, 187, 201] [49, 62, 65, 79, 181, 188, 202, 204] [11, 23, 63, 122, 156, 177, 189, 247] [12, 24, 64, 81, 157, 178, 190, 248] [13, 65, 82, 93, 158, 179, 191, 207] [26, 66, 92, 123, 142, 159, 192, 248] [27, 36, 67, 84, 160, 193, 207, 249] [0, 37, 68, 85, 161, 194, 208, 210] [19, 54, 69, 95, 144, 195, 240, 249] [39, 55, 70, 87, 145, 189, 196, 210] [40, 56, 71, 88, 190, 197, 211, 213] [33, 49, 72, 124, 147, 165, 186, 198] [34, 50, 73, 90, 166, 187, 199, 250] [5, 35, 74, 91, 167, 188, 200, 216] [24, 52, 75, 114, 149, 168, 201, 250] [53, 63, 76, 93, 150, 169, 202, 216] [6, 64, 77, 94, 170, 203, 217, 219] [27, 78, 81, 116, 150, 152, 204, 245] [0, 66, 79, 82, 153, 198, 205, 219] [67, 80, 83, 96, 126, 199, 206, 220] [25, 40, 81, 125, 155, 174, 195, 207] [26, 41, 82, 98, 175, 196, 208, 251] [2, 27, 83, 99, 176, 197, 209, 224] [32, 43, 84, 108, 157, 177, 210, 251] [44, 54, 85, 101, 158, 178, 211, 224] [3, 55, 86, 102, 179, 212, 225, 227] [35, 72, 87, 110, 158, 160, 213, 248] [1, 57, 73, 88, 161, 207, 214, 227] [58, 74, 89, 104, 127, 208, 215, 228] [38, 51, 67, 90, 163, 183, 204, 216] [52, 68, 91, 106, 164, 184, 205, 217] [15, 53, 92, 107, 185, 206, 218, 232] [41, 70, 93, 122, 164, 166, 186, 219] [2, 71, 81, 94, 167, 187, 220, 232] [17, 82, 95, 109, 128, 188, 221, 233] [3, 84, 96, 99, 128, 170, 216, 222] [4, 85, 97, 100, 129, 217, 223, 235] [42, 47, 58, 98, 172, 192, 213, 224] [43, 59, 99, 112, 173, 193, 214, 225] [9, 44, 100, 113, 194, 215, 226, 238] [50, 61, 101, 119, 173, 175, 195, 227] [5, 62, 72, 102, 176, 196, 228, 238] [11, 73, 103, 115, 131, 197, 229, 239] [6, 75, 91, 104, 131, 179, 224, 230] [7, 76, 92, 105, 132, 225, 231, 241] [56, 69, 85, 106, 126, 181, 201, 232] [8, 70, 86, 107, 182, 202, 222, 233] [30, 71, 108, 118, 134, 203, 223, 234] [9, 89, 98, 109, 134, 185, 205, 235] [10, 33, 99, 110, 135, 206, 236, 244] [12, 102, 111, 114, 136, 137, 233, 237] [60, 65, 76, 112, 127, 190, 210, 238] [14, 61, 77, 113, 191, 211, 230, 239] [22, 62, 114, 121, 140, 212, 231, 240] [15, 80, 90, 115, 140, 194, 214, 241] [16, 25, 91, 116, 141, 215, 242, 247] [18, 94, 108, 117, 142, 143, 239, 243] [20, 88, 103, 118, 130, 200, 220, 244] [21, 48, 89, 119, 146, 221, 237, 245] [23, 51, 113, 120, 147, 148, 223, 246] [28, 79, 95, 121, 133, 209, 228, 247] [29, 39, 80, 122, 154, 229, 243, 248] [31, 42, 107, 123, 155, 156, 231, 249] [37, 66, 105, 124, 139, 162, 236, 250] [46, 57, 97, 125, 145, 171, 242, 251]
H_Z (126 checks, sparse supports)
[0, 9, 80, 106, 126, 154, 194, 205] [1, 15, 89, 112, 127, 146, 185, 214] [2, 20, 95, 96, 128, 144, 209, 220] [3, 8, 22, 97, 129, 171, 212, 222] [4, 23, 42, 118, 130, 156, 172, 223] [5, 28, 103, 104, 131, 138, 200, 228] [6, 14, 30, 105, 132, 162, 203, 230] [7, 31, 51, 121, 133, 148, 163, 231] [8, 12, 108, 109, 134, 143, 157, 233] [4, 9, 36, 110, 135, 160, 226, 235] [10, 37, 57, 111, 136, 161, 171, 236] [11, 20, 39, 111, 130, 137, 189, 229] [12, 21, 40, 60, 138, 174, 190, 237] [13, 41, 61, 124, 139, 175, 186, 191] [14, 18, 114, 115, 137, 140, 149, 239] [7, 15, 45, 116, 141, 152, 218, 241] [16, 46, 66, 117, 142, 153, 162, 242] [17, 28, 48, 117, 133, 143, 180, 221] [18, 29, 49, 69, 144, 165, 181, 243] [19, 50, 70, 125, 145, 166, 182, 195] [10, 20, 26, 119, 146, 159, 175, 244] [21, 27, 72, 120, 147, 160, 176, 245] [12, 22, 54, 120, 136, 148, 178, 240] [13, 23, 55, 75, 149, 179, 189, 246] [24, 56, 76, 78, 132, 150, 190, 201] [13, 25, 36, 57, 138, 151, 207, 242] [26, 37, 58, 78, 139, 152, 192, 208] [27, 38, 59, 79, 153, 193, 204, 209] [16, 28, 34, 122, 151, 154, 166, 247] [29, 35, 81, 123, 152, 155, 167, 248] [18, 30, 63, 123, 142, 156, 169, 234] [19, 31, 64, 84, 157, 170, 180, 249] [32, 65, 85, 87, 129, 158, 181, 210] [19, 33, 45, 66, 144, 159, 198, 236] [34, 46, 67, 87, 145, 160, 183, 199] [35, 47, 68, 88, 161, 184, 200, 213] [23, 36, 43, 124, 147, 162, 177, 193] [24, 37, 44, 90, 163, 178, 194, 250] [3, 38, 91, 93, 141, 164, 179, 216] [24, 26, 39, 72, 149, 165, 196, 248] [27, 40, 73, 93, 150, 166, 197, 207] [0, 41, 74, 94, 143, 167, 208, 219] [27, 42, 54, 75, 152, 168, 224, 249] [0, 43, 55, 76, 153, 169, 210, 225] [44, 56, 77, 96, 126, 170, 211, 226] [31, 45, 52, 125, 155, 168, 171, 184] [32, 46, 53, 98, 169, 172, 185, 251] [6, 47, 99, 101, 135, 170, 173, 224] [32, 34, 48, 81, 157, 174, 187, 245] [35, 49, 82, 101, 158, 175, 188, 198] [1, 50, 83, 102, 137, 176, 199, 227] [35, 51, 63, 84, 160, 177, 216, 246] [1, 52, 64, 85, 161, 178, 201, 217] [53, 65, 86, 104, 127, 179, 202, 218] [38, 40, 54, 61, 163, 180, 195, 211] [41, 55, 62, 106, 164, 181, 196, 212] [2, 11, 56, 107, 156, 182, 197, 232] [41, 43, 57, 90, 166, 183, 214, 251] [2, 44, 58, 91, 167, 184, 215, 224] [3, 59, 92, 109, 128, 159, 185, 225] [44, 60, 72, 93, 158, 169, 186, 238] [3, 61, 73, 94, 170, 187, 227, 239] [4, 62, 74, 95, 129, 188, 228, 240] [47, 49, 63, 70, 172, 186, 189, 202] [50, 64, 71, 112, 173, 187, 190, 203] [5, 17, 65, 113, 148, 188, 191, 238] [50, 52, 66, 98, 175, 192, 205, 250] [5, 53, 67, 99, 176, 193, 206, 216] [6, 68, 100, 115, 131, 151, 194, 217] [53, 69, 81, 101, 150, 178, 195, 232] [6, 70, 82, 102, 179, 196, 219, 233] [7, 71, 83, 103, 132, 197, 220, 234] [56, 58, 72, 79, 181, 198, 213, 228] [8, 59, 73, 80, 182, 199, 214, 229] [9, 25, 74, 118, 134, 174, 200, 215] [59, 61, 75, 106, 173, 184, 201, 230] [9, 62, 76, 107, 185, 202, 231, 238] [10, 11, 77, 108, 135, 177, 203, 239] [2, 62, 78, 90, 140, 176, 187, 204] [11, 79, 91, 109, 131, 188, 205, 247] [12, 80, 92, 110, 137, 206, 241, 248] [65, 67, 81, 88, 190, 204, 207, 220] [14, 68, 82, 89, 191, 205, 208, 221] [15, 33, 83, 121, 140, 165, 206, 209] [68, 70, 84, 112, 164, 193, 210, 222] [15, 71, 85, 113, 194, 211, 223, 232] [16, 17, 86, 114, 141, 168, 212, 233] [5, 71, 87, 98, 134, 167, 196, 213] [17, 88, 99, 115, 128, 197, 214, 244] [18, 89, 100, 116, 143, 215, 235, 245] [74, 76, 90, 96, 127, 199, 216, 241] [20, 77, 91, 97, 200, 217, 230, 242] [21, 22, 42, 92, 146, 192, 218, 231] [8, 77, 79, 93, 133, 191, 202, 219] [22, 80, 94, 118, 140, 203, 220, 243] [23, 25, 95, 119, 148, 195, 221, 247] [10, 25, 96, 107, 141, 155, 206, 222] [26, 97, 108, 120, 142, 151, 223, 251] [83, 85, 98, 104, 126, 208, 224, 235] [28, 86, 99, 105, 209, 222, 225, 236] [29, 30, 51, 100, 154, 183, 223, 226] [14, 86, 88, 101, 130, 182, 211, 227] [30, 89, 102, 121, 134, 212, 228, 237] [31, 33, 103, 122, 156, 186, 229, 244] [16, 33, 104, 113, 135, 147, 215, 230] [34, 105, 114, 123, 136, 159, 231, 250] [4, 92, 94, 106, 132, 142, 217, 232] [36, 95, 107, 111, 133, 218, 233, 249] [37, 39, 60, 108, 162, 210, 234, 243] [21, 39, 97, 109, 145, 154, 221, 235] [40, 42, 110, 124, 155, 165, 213, 236] [24, 43, 111, 119, 157, 168, 173, 237] [7, 100, 102, 112, 129, 136, 225, 238] [45, 103, 113, 117, 130, 226, 239, 246] [46, 48, 69, 114, 171, 201, 237, 240] [29, 48, 105, 115, 139, 146, 229, 241] [49, 51, 116, 125, 147, 174, 204, 242] [32, 52, 117, 122, 149, 164, 177, 243] [13, 54, 110, 118, 144, 158, 234, 244] [55, 57, 78, 119, 145, 180, 227, 245] [38, 58, 60, 120, 127, 172, 183, 246] [19, 63, 116, 121, 138, 150, 240, 247] [64, 66, 87, 122, 139, 189, 219, 248] [47, 67, 69, 123, 126, 163, 192, 249] [0, 73, 75, 124, 131, 161, 198, 250] [1, 82, 84, 125, 128, 153, 207, 251]
Code ID 252-22-19 · download JSON · raw on GitHub