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[[288,6,16]] d ≤
n
288
k
6
d
16
kd²/n
5.333
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 16 · witness weight 16 (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, 16 qubits)
[20, 21, 33, 37, 69, 83, 93, 102, 158, 164, 199, 213, 246, 257, 262, 274]
d_Z 16 · witness weight 16 (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, 16 qubits)
[4, 25, 36, 42, 60, 68, 84, 123, 169, 180, 187, 194, 211, 235, 256, 267]
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 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×144 (2,2)×144 (3,2)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 144 (1,3): 144 (2,2): 144 (2,3): 864 (2,4): 432 (3,2): 144 (3,3): 2640 (3,4): 4752 (3,5): 2016 (3,6): 864 (3,7): 144
trapping sets H_Z (1,2)×144 (2,2)×144 (3,2)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 144 (1,3): 144 (2,2): 144 (2,3): 864 (2,4): 432 (3,2): 144 (3,3): 2640 (3,4): 4752 (3,5): 2016 (3,6): 864 (3,7): 144

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C3 x (C3 : C16) = SmallGroup(144, 28) (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, 7, 39}, b = {0, 10} as element indices. Check weight 5. 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

[[288,6,16]] — two-block group-algebra code over C3 x (C3 : C16)

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/288-6-16.json.

Parity checks

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