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[[288,8,22]] d ≤
n
288
k
8
d
22
kd²/n
13.444
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 22, d_Z ≤ 22 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 22 · witness weight 22 (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, 22 qubits)
[13, 16, 26, 36, 70, 72, 80, 102, 118, 129, 140, 150, 162, 179, 181, 196, 200, 206, 214, 222, 226, 241]
d_Z 22 · witness weight 22 (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, 22 qubits)
[24, 27, 41, 44, 51, 56, 58, 77, 92, 100, 130, 148, 152, 160, 167, 191, 203, 204, 240, 248, 250, 273]
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 6 · H_Z 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×288 (2,4)×2160 (3,5)×21600 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 288 (2,4): 2160 (3,5): 21600 (3,7): 2880
trapping sets H_Z (1,3)×288 (2,4)×2160 (3,5)×21600 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 288 (2,4): 2160 (3,5): 21600 (3,7): 2880

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C18 x D8 = SmallGroup(144, 48) (order 144), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 288; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 9, 39}, b = {0, 31, 74} as element indices. Check weight 6. 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

[[288,8,22]] — two-block group-algebra code over C18 x D8

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| = 288

Parameters are also spelled out in provenance.construction of codes/288-8-22.json.

Parity checks

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