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[[300,32,15]] d ≤
n
300
k
32
d
15
kd²/n
24.0
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (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, 15 qubits)
[200, 225, 226, 250, 251, 252, 271, 272, 273, 286, 287, 288, 295, 296, 299]
d_Z 15 · witness weight 15 (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, 15 qubits)
[16, 37, 50, 79, 108, 160, 164, 170, 184, 199, 211, 219, 226, 240, 255]
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)×300 (2,6)×4200 (3,6)×2100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 300 (2,6): 4200 (3,6): 2100 (3,8): 81900 (3,10): 8400
trapping sets H_Z (1,4)×300 (2,6)×4200 (3,6)×2100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 300 (2,6): 4200 (3,6): 2100 (3,8): 81900 (3,10): 8400

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C5 x D30 = SmallGroup(150, 11) (order 150), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 300; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 21, 48, 69}, b = {0, 5, 106, 142} 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 2bga-nonabelian (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

[[300,32,15]] — two-block group-algebra code over C5 x D30

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| = 300

Parameters are also spelled out in provenance.construction of codes/300-32-15.json.

Parity checks

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