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[[264,8,11]] d ≤
n
264
k
8
d
11
kd²/n
3.667
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · 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 11 · witness weight 11 (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, 11 qubits)
[10, 21, 33, 45, 57, 69, 81, 93, 105, 117, 126]
d_Z 11 · witness weight 11 (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, 11 qubits)
[135, 143, 154, 166, 178, 190, 202, 214, 226, 238, 250]
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)×132 (2,2)×132 (3,2)×132 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 132 (1,3): 132 (2,2): 132 (2,3): 792 (2,4): 396 (3,2): 132 (3,3): 2508 (3,4): 4356 (3,5): 1584 (3,6): 792 (3,7): 132
trapping sets H_Z (1,2)×132 (2,2)×132 (3,2)×132 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 132 (1,3): 132 (2,2): 132 (2,3): 792 (2,4): 396 (3,2): 132 (3,3): 2508 (3,4): 4356 (3,5): 1584 (3,6): 792 (3,7): 132

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C66 x C2 = SmallGroup(132, 10) (order 132), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 264; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 6, 30}, b = {0, 23} 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 generalized bicycle (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

[[264,8,11]] — two-block group-algebra code over C66 x C2

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| = 264

Parameters are also spelled out in provenance.construction of codes/264-8-11.json.

Parity checks

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