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[[306,26,19]] d ≤
n
306
k
26
d
19
kd²/n
30.673
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 5000 trials · survived 5000 trials · 2026-10-12
witness operator (support, 19 qubits)
[4, 13, 33, 38, 49, 68, 69, 74, 94, 105, 130, 141, 166, 202, 227, 246, 258, 294, 303]
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)
[7, 31, 67, 79, 92, 128, 140, 160, 164, 165, 184, 195, 200, 220, 231, 245, 256, 281, 292]
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)×306 (2,6)×4284 (3,6)×1530 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 306 (2,6): 4284 (3,6): 1530 (3,8): 85374 (3,10): 8568
trapping sets H_Z (1,4)×306 (2,6)×4284 (3,6)×1530 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 306 (2,6): 4284 (3,6): 1530 (3,8): 85374 (3,10): 8568

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C51 x C3 = SmallGroup(153, 2) (order 153), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 306; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 32, 91, 124}, b = {0, 19, 31, 56} 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. The X witness is the database's own published logical operator rather than one this session's search produced; that side's witness_provenance records the difference. 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

[[306,26,19]] — two-block group-algebra code over C51 x C3

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 5,000 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| = 306

Parameters are also spelled out in provenance.construction of codes/306-26-19.json.

Parity checks

X-checks 153 (max weight 8) · Z-checks 153 (max weight 8)
H_X (153 checks, sparse supports)
[0, 33, 84, 119, 153, 248, 296, 304] [1, 39, 90, 107, 154, 254, 285, 305] [2, 17, 68, 127, 155, 256, 281, 293] [3, 42, 93, 128, 156, 165, 257, 302] [4, 45, 78, 113, 157, 242, 291, 301] [5, 23, 74, 115, 158, 262, 269, 298] [6, 48, 99, 116, 159, 171, 263, 294] [7, 25, 76, 135, 160, 264, 289, 300] [8, 26, 77, 136, 154, 161, 265, 290] [9, 51, 102, 137, 162, 174, 266, 305] [10, 29, 62, 121, 163, 250, 275, 287] [11, 54, 87, 122, 160, 164, 251, 299] [12, 31, 82, 123, 165, 270, 277, 303] [13, 32, 83, 124, 157, 166, 271, 278] [14, 57, 108, 125, 167, 180, 272, 301] [15, 34, 85, 143, 158, 168, 273, 297] [16, 35, 86, 144, 159, 169, 274, 298] [17, 60, 111, 145, 170, 171, 183, 275] [18, 37, 70, 129, 171, 258, 283, 295] [19, 38, 71, 130, 153, 172, 259, 284] [20, 63, 96, 131, 168, 173, 260, 304] [21, 40, 91, 132, 163, 174, 279, 286] [22, 41, 92, 133, 164, 175, 280, 287] [23, 66, 117, 134, 160, 176, 189, 281] [24, 43, 94, 149, 166, 177, 282, 303] [25, 44, 95, 150, 157, 167, 178, 283] [4, 26, 69, 120, 179, 180, 192, 284] [27, 46, 79, 138, 155, 180, 267, 292] [28, 47, 80, 139, 156, 181, 268, 293] [29, 72, 105, 140, 165, 177, 182, 269] [30, 49, 100, 141, 172, 183, 288, 295] [31, 50, 101, 142, 153, 173, 184, 289] [0, 32, 75, 126, 168, 185, 198, 290] [33, 52, 103, 152, 163, 175, 186, 291] [10, 34, 53, 104, 164, 176, 187, 292] [11, 35, 78, 129, 188, 189, 201, 293] [36, 55, 88, 146, 161, 189, 276, 300] [37, 56, 89, 147, 154, 162, 190, 277] [1, 38, 81, 114, 174, 186, 191, 278] [39, 58, 109, 148, 155, 181, 192, 296] [2, 40, 59, 110, 156, 182, 193, 297] [3, 41, 84, 135, 177, 194, 207, 298] [18, 42, 61, 112, 172, 184, 195, 299] [19, 43, 62, 113, 173, 185, 196, 300] [20, 44, 87, 138, 154, 197, 198, 210] [45, 64, 97, 151, 158, 169, 198, 285] [5, 46, 65, 98, 159, 170, 199, 286] [6, 47, 90, 123, 183, 195, 200, 287] [7, 48, 67, 118, 161, 190, 201, 302] [8, 49, 68, 119, 162, 191, 202, 303] [9, 50, 93, 143, 157, 186, 203, 216] [27, 51, 70, 121, 181, 193, 204, 304] [28, 52, 71, 122, 158, 182, 194, 205] [29, 53, 96, 146, 159, 206, 207, 219] [12, 54, 73, 106, 166, 178, 207, 294] [13, 55, 74, 107, 167, 179, 208, 295] [14, 56, 99, 132, 153, 192, 204, 209] [15, 57, 76, 127, 169, 199, 210, 305] [16, 58, 77, 128, 163, 170, 200, 211] [17, 59, 102, 149, 164, 195, 212, 225] [36, 60, 79, 130, 165, 190, 202, 213] [37, 61, 80, 131, 166, 191, 203, 214] [38, 62, 105, 151, 167, 215, 216, 228] [21, 63, 82, 115, 175, 187, 216, 301] [22, 64, 83, 116, 155, 176, 188, 217] [23, 65, 108, 141, 156, 201, 213, 218] [24, 66, 85, 136, 171, 178, 208, 219] [25, 67, 86, 137, 172, 179, 209, 220] [26, 68, 111, 152, 173, 204, 221, 234] [45, 69, 88, 139, 174, 199, 211, 222] [46, 70, 89, 140, 175, 200, 212, 223] [12, 47, 71, 114, 176, 224, 225, 237] [30, 72, 91, 124, 160, 184, 196, 225] [31, 73, 92, 125, 161, 185, 197, 226] [32, 74, 117, 148, 162, 210, 222, 227] [33, 75, 94, 144, 180, 187, 217, 228] [34, 76, 95, 145, 181, 188, 218, 229] [18, 35, 77, 120, 182, 213, 230, 243] [54, 78, 97, 147, 183, 208, 220, 231] [1, 55, 79, 98, 184, 209, 221, 232] [21, 56, 80, 123, 185, 233, 234, 246] [39, 81, 100, 133, 168, 193, 205, 234] [40, 82, 101, 134, 169, 194, 206, 235] [7, 41, 83, 126, 170, 219, 231, 236] [42, 84, 103, 150, 189, 196, 226, 237] [4, 43, 85, 104, 190, 197, 227, 238] [27, 44, 86, 129, 191, 222, 239, 252] [5, 63, 87, 106, 192, 217, 229, 240] [6, 64, 88, 107, 193, 218, 230, 241] [30, 65, 89, 132, 194, 242, 243, 255] [48, 90, 109, 142, 177, 202, 214, 243] [0, 49, 91, 110, 178, 203, 215, 244] [15, 50, 92, 135, 179, 228, 240, 245] [10, 51, 93, 112, 198, 205, 235, 246] [11, 52, 94, 113, 199, 206, 236, 247] [36, 53, 95, 138, 200, 231, 248, 261] [13, 72, 96, 115, 201, 226, 238, 249] [14, 73, 97, 116, 202, 227, 239, 250] [39, 74, 98, 141, 203, 251, 252, 264] [2, 57, 99, 118, 186, 211, 223, 252] [3, 58, 100, 119, 187, 212, 224, 253] [24, 59, 101, 143, 188, 237, 249, 254] [19, 60, 102, 121, 207, 214, 244, 255] [20, 61, 103, 122, 208, 215, 245, 256] [45, 62, 104, 146, 209, 240, 257, 270] [22, 81, 105, 124, 210, 235, 247, 258] [23, 82, 106, 125, 211, 236, 248, 259] [48, 83, 107, 148, 212, 260, 261, 273] [8, 66, 108, 127, 195, 220, 232, 261] [9, 67, 109, 128, 196, 221, 233, 262] [33, 68, 110, 149, 197, 246, 258, 263] [28, 69, 111, 130, 216, 223, 253, 264] [29, 70, 112, 131, 217, 224, 254, 265] [54, 71, 113, 151, 218, 249, 266, 279] [31, 90, 114, 133, 219, 244, 256, 267] [32, 91, 115, 134, 220, 245, 257, 268] [7, 57, 92, 116, 221, 269, 270, 282] [16, 75, 117, 136, 204, 229, 241, 270] [17, 76, 118, 137, 205, 230, 242, 271] [42, 77, 119, 152, 206, 255, 267, 272] [37, 78, 120, 139, 225, 232, 262, 273] [38, 79, 121, 140, 226, 233, 263, 274] [12, 63, 80, 122, 227, 258, 275, 288] [40, 99, 123, 142, 228, 253, 265, 276] [0, 41, 100, 124, 229, 254, 266, 277] [15, 66, 101, 125, 230, 278, 279, 291] [25, 84, 126, 144, 213, 238, 250, 279] [26, 85, 127, 145, 214, 239, 251, 280] [18, 51, 86, 128, 215, 264, 276, 281] [46, 87, 129, 147, 234, 241, 271, 282] [1, 47, 88, 130, 235, 242, 272, 283] [21, 72, 89, 131, 236, 267, 284, 296] [2, 49, 108, 132, 237, 262, 274, 285] [3, 50, 109, 133, 238, 263, 275, 286] [24, 75, 110, 134, 239, 287, 288, 299] [34, 93, 135, 150, 222, 247, 259, 288] [4, 35, 94, 136, 223, 248, 260, 289] [27, 60, 95, 137, 224, 273, 285, 290] [5, 55, 96, 138, 243, 250, 280, 291] [6, 56, 97, 139, 244, 251, 281, 292] [30, 81, 98, 140, 245, 276, 293, 302] [8, 58, 117, 141, 246, 271, 283, 294] [9, 59, 118, 142, 247, 272, 284, 295] [10, 43, 102, 143, 231, 256, 268, 296] [11, 44, 103, 144, 232, 257, 269, 297] [36, 69, 104, 145, 233, 282, 294, 298] [13, 64, 105, 146, 252, 259, 289, 299] [14, 65, 106, 147, 253, 260, 290, 300] [16, 67, 126, 148, 255, 280, 292, 301] [19, 52, 111, 149, 240, 265, 277, 302] [20, 53, 112, 150, 241, 266, 278, 303] [22, 73, 114, 151, 261, 268, 297, 304] [28, 61, 120, 152, 249, 274, 286, 305]
H_Z (153 checks, sparse supports)
[0, 19, 31, 56, 153, 185, 244, 277] [1, 8, 37, 44, 154, 191, 232, 283] [2, 27, 39, 64, 155, 193, 252, 285] [3, 28, 40, 65, 156, 194, 253, 286] [4, 13, 25, 50, 157, 179, 238, 289] [5, 15, 45, 52, 158, 199, 240, 291] [6, 16, 46, 53, 159, 200, 241, 292] [7, 11, 23, 72, 160, 201, 236, 269] [8, 36, 48, 73, 161, 202, 261, 294] [9, 37, 49, 74, 162, 203, 262, 295] [10, 21, 33, 58, 163, 187, 246, 296] [11, 22, 34, 59, 164, 188, 247, 297] [3, 12, 29, 60, 165, 207, 224, 275] [13, 24, 54, 61, 166, 208, 249, 299] [14, 25, 55, 62, 167, 209, 250, 300] [15, 20, 32, 81, 168, 210, 245, 278] [16, 45, 57, 82, 169, 211, 270, 301] [17, 46, 58, 83, 155, 170, 212, 271] [6, 17, 18, 66, 171, 195, 230, 281] [19, 30, 42, 67, 172, 196, 255, 302] [20, 31, 43, 68, 173, 197, 256, 303] [9, 21, 38, 69, 174, 216, 233, 284] [22, 33, 63, 70, 175, 217, 258, 304] [23, 34, 64, 71, 158, 176, 218, 259] [24, 29, 41, 90, 177, 219, 254, 287] [25, 54, 66, 91, 160, 178, 220, 279] [26, 55, 67, 92, 161, 179, 221, 280] [14, 26, 27, 75, 180, 204, 239, 290] [28, 39, 51, 76, 181, 205, 264, 305] [29, 40, 52, 77, 163, 182, 206, 265] [17, 30, 47, 78, 183, 225, 242, 293] [31, 42, 72, 79, 165, 184, 226, 267] [32, 43, 73, 80, 166, 185, 227, 268] [33, 38, 50, 99, 153, 186, 228, 263] [34, 63, 75, 100, 168, 187, 229, 288] [35, 64, 76, 101, 169, 188, 230, 289] [23, 35, 36, 84, 189, 213, 248, 298] [37, 48, 60, 85, 171, 190, 214, 273] [38, 49, 61, 86, 172, 191, 215, 274] [26, 39, 56, 87, 154, 192, 234, 251] [40, 51, 81, 88, 174, 193, 235, 276] [41, 52, 82, 89, 175, 194, 236, 277] [42, 47, 59, 108, 156, 195, 237, 272] [43, 72, 84, 109, 177, 196, 238, 296] [44, 73, 85, 110, 178, 197, 239, 297] [32, 44, 45, 93, 157, 198, 222, 257] [46, 57, 69, 94, 180, 199, 223, 282] [47, 58, 70, 95, 181, 200, 224, 283] [35, 48, 65, 96, 159, 201, 243, 260] [49, 60, 90, 97, 183, 202, 244, 285] [50, 61, 91, 98, 184, 203, 245, 286] [51, 56, 68, 117, 162, 204, 246, 281] [52, 81, 93, 118, 186, 205, 247, 302] [53, 82, 94, 119, 187, 206, 248, 303] [41, 53, 54, 102, 164, 207, 231, 266] [55, 66, 78, 103, 189, 208, 232, 291] [56, 67, 79, 104, 190, 209, 233, 292] [44, 57, 74, 105, 167, 210, 252, 269] [58, 69, 99, 106, 192, 211, 253, 294] [59, 70, 100, 107, 193, 212, 254, 295] [60, 65, 77, 126, 170, 213, 255, 290] [61, 90, 102, 127, 195, 214, 256, 305] [62, 91, 103, 128, 163, 196, 215, 257] [50, 62, 63, 111, 173, 216, 240, 275] [64, 75, 87, 112, 198, 217, 241, 299] [65, 76, 88, 113, 199, 218, 242, 300] [53, 66, 83, 114, 176, 219, 261, 278] [67, 78, 108, 115, 201, 220, 262, 301] [68, 79, 109, 116, 155, 202, 221, 263] [69, 74, 86, 135, 179, 222, 264, 298] [70, 99, 111, 136, 171, 204, 223, 265] [71, 100, 112, 137, 172, 205, 224, 266] [59, 71, 72, 120, 182, 225, 249, 284] [73, 84, 96, 121, 207, 226, 250, 304] [74, 85, 97, 122, 158, 208, 227, 251] [62, 75, 92, 123, 185, 228, 270, 287] [76, 87, 117, 124, 160, 210, 229, 271] [77, 88, 118, 125, 161, 211, 230, 272] [78, 83, 95, 143, 157, 188, 231, 273] [79, 108, 120, 144, 180, 213, 232, 274] [80, 109, 121, 145, 181, 214, 233, 275] [68, 80, 81, 129, 191, 234, 258, 293] [82, 93, 105, 130, 165, 216, 235, 259] [83, 94, 106, 131, 166, 217, 236, 260] [71, 84, 101, 132, 153, 194, 237, 279] [85, 96, 126, 133, 168, 219, 238, 280] [86, 97, 127, 134, 169, 220, 239, 281] [87, 92, 104, 149, 164, 197, 240, 282] [88, 117, 129, 150, 189, 222, 241, 283] [4, 89, 118, 130, 190, 223, 242, 284] [77, 89, 90, 138, 154, 200, 243, 267] [91, 102, 114, 139, 174, 225, 244, 268] [92, 103, 115, 140, 175, 226, 245, 269] [80, 93, 110, 141, 156, 203, 246, 288] [94, 105, 135, 142, 177, 228, 247, 289] [0, 95, 106, 136, 178, 229, 248, 290] [96, 101, 113, 152, 173, 206, 249, 291] [10, 97, 126, 138, 198, 231, 250, 292] [11, 98, 127, 139, 199, 232, 251, 293] [86, 98, 99, 146, 159, 209, 252, 276] [100, 111, 123, 147, 183, 234, 253, 277] [1, 101, 112, 124, 184, 235, 254, 278] [89, 102, 119, 148, 162, 212, 255, 296] [2, 103, 114, 143, 186, 237, 256, 297] [3, 104, 115, 144, 187, 238, 257, 298] [18, 105, 110, 122, 182, 215, 258, 299] [19, 106, 135, 146, 207, 240, 259, 300] [20, 107, 136, 147, 154, 208, 241, 260] [95, 107, 108, 151, 167, 218, 261, 285] [5, 109, 120, 132, 192, 243, 262, 286] [6, 110, 121, 133, 193, 244, 263, 287] [7, 98, 111, 128, 170, 221, 264, 302] [8, 112, 123, 149, 195, 246, 265, 303] [9, 113, 124, 150, 157, 196, 247, 266] [27, 114, 119, 131, 191, 224, 267, 304] [28, 115, 143, 151, 158, 216, 249, 268] [5, 29, 116, 144, 159, 217, 250, 269] [12, 104, 116, 117, 176, 227, 270, 294] [13, 118, 129, 141, 201, 252, 271, 295] [14, 119, 130, 142, 153, 202, 253, 272] [15, 107, 120, 137, 179, 230, 273, 305] [16, 121, 132, 152, 163, 204, 255, 274] [10, 17, 122, 133, 164, 205, 256, 275] [36, 123, 128, 140, 165, 200, 233, 276] [12, 37, 124, 149, 166, 225, 258, 277] [13, 38, 125, 150, 167, 226, 259, 278] [21, 113, 125, 126, 185, 236, 279, 301] [22, 127, 138, 148, 155, 210, 261, 280] [2, 23, 128, 139, 156, 211, 262, 281] [24, 116, 129, 145, 171, 188, 239, 282] [18, 25, 130, 141, 172, 213, 264, 283] [19, 26, 131, 142, 173, 214, 265, 284] [1, 45, 132, 137, 174, 209, 242, 285] [21, 46, 133, 152, 175, 234, 267, 286] [10, 22, 47, 134, 176, 235, 268, 287] [30, 122, 134, 135, 160, 194, 245, 288] [7, 31, 136, 146, 161, 219, 270, 289] [8, 32, 137, 147, 162, 220, 271, 290] [4, 33, 125, 138, 180, 197, 248, 291] [27, 34, 139, 148, 181, 222, 273, 292] [2, 28, 35, 140, 182, 223, 274, 293] [6, 54, 141, 145, 183, 218, 251, 294] [18, 30, 55, 142, 184, 243, 276, 295] [0, 39, 131, 143, 168, 203, 254, 296] [15, 40, 144, 151, 169, 228, 279, 297] [5, 16, 41, 145, 170, 229, 280, 298] [11, 42, 134, 146, 189, 206, 257, 299] [7, 36, 43, 147, 190, 231, 282, 300] [4, 14, 63, 148, 192, 227, 260, 301] [3, 48, 140, 149, 177, 212, 263, 302] [12, 24, 49, 150, 178, 237, 288, 303] [0, 20, 51, 151, 198, 215, 266, 304] [1, 9, 57, 152, 186, 221, 272, 305]
Code ID 306-26-19 · download JSON · raw on GitHub