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[[300,152,2]] d =
n
300
k
152
d
2
kd²/n
2.027
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (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, 2 qubits)
[120, 129]
d_Z 2 · witness weight 2 (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, 2 qubits)
[88, 100]
certificate exact, d = 2 · CryptoMiniSat 5.14.7 SAT
X: no logical < 2 exists; Z: no logical < 2 exists

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 4 · H_Z 4 (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 (2,0)×150 (3,4)×12600 (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 (2,0): 150 (2,4): 1800 (3,4): 12600 (3,8): 2400
trapping sets H_Z (1,4)×300 (2,0)×150 (3,4)×12600 (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 (2,0): 150 (2,4): 1800 (3,4): 12600 (3,8): 2400

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C30 x C5 = SmallGroup(150, 13) (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, 1, 40, 52}, b = {0, 1, 12, 18} 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

[[300,152,2]] — two-block group-algebra code over C30 x C5

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/300-152-2.json.

Parity checks

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