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[[200,8,10]] d ≤
n
200
k
8
d
10
kd²/n
4.0
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · 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 10 · witness weight 10 (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, 10 qubits)
[0, 21, 50, 63, 76, 90, 100, 145, 150, 163]
d_Z 10 · witness weight 10 (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, 10 qubits)
[39, 49, 58, 81, 113, 129, 139, 149, 176, 184]
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 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)×100 (2,2)×100 (3,2)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 100 (1,3): 100 (2,2): 100 (2,3): 600 (2,4): 300 (3,2): 100 (3,3): 1800 (3,4): 3300 (3,5): 1500 (3,6): 600 (3,7): 100
trapping sets H_Z (1,2)×100 (2,2)×100 (3,2)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 100 (1,3): 100 (2,2): 100 (2,3): 600 (2,4): 300 (3,2): 100 (3,3): 1800 (3,4): 3300 (3,5): 1500 (3,6): 600 (3,7): 100

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C10 x D10 = SmallGroup(100, 14) (order 100), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 200; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 6, 22}, b = {0, 19} 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

[[200,8,10]] — two-block group-algebra code over C10 x D10

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| = 200

Parameters are also spelled out in provenance.construction of codes/200-8-10.json.

Parity checks

X-checks 100 (max weight 5) · Z-checks 100 (max weight 5)
H_X (100 checks, sparse supports)
[0, 45, 86, 100, 198] [1, 36, 46, 101, 199] [2, 59, 93, 102, 194] [1, 3, 94, 103, 154] [4, 21, 64, 104, 150] [5, 50, 60, 105, 197] [0, 6, 64, 106, 163] [7, 55, 65, 107, 159] [5, 8, 98, 108, 140] [9, 32, 75, 109, 136] [6, 10, 40, 110, 171] [7, 11, 36, 102, 111] [12, 37, 79, 112, 168] [2, 13, 75, 113, 149] [14, 68, 76, 114, 145] [3, 7, 15, 115, 178] [4, 16, 79, 105, 116] [17, 72, 80, 117, 175] [13, 18, 54, 118, 158] [14, 19, 50, 100, 119] [20, 51, 87, 120, 155] [15, 21, 58, 121, 184] [0, 16, 22, 108, 122] [17, 23, 55, 109, 123] [24, 56, 90, 124, 182] [8, 14, 25, 125, 167] [9, 26, 87, 101, 126] [27, 82, 88, 127, 164] [10, 16, 28, 128, 189] [11, 17, 29, 113, 129] [12, 30, 90, 114, 130] [31, 85, 91, 131, 187] [25, 32, 71, 132, 174] [2, 26, 33, 103, 133] [27, 34, 68, 104, 134] [35, 69, 95, 135, 172] [28, 36, 74, 136, 193] [3, 29, 37, 118, 137] [4, 30, 38, 119, 138] [31, 39, 72, 120, 139] [40, 73, 97, 140, 192] [18, 26, 41, 141, 181] [19, 27, 42, 106, 142] [20, 43, 95, 107, 143] [44, 92, 96, 144, 179] [21, 29, 45, 145, 196] [22, 30, 46, 125, 146] [23, 31, 47, 126, 147] [24, 48, 97, 127, 148] [28, 49, 94, 149, 195] [41, 50, 84, 150, 186] [8, 42, 51, 110, 151] [9, 43, 52, 111, 152] [44, 53, 82, 112, 153] [54, 83, 99, 154, 185] [10, 46, 55, 132, 155] [11, 47, 56, 133, 156] [12, 48, 57, 134, 157] [49, 58, 85, 135, 158] [32, 42, 59, 159, 191] [33, 43, 60, 115, 160] [34, 44, 61, 116, 161] [35, 62, 99, 117, 162] [41, 63, 98, 163, 190] [37, 47, 64, 141, 164] [38, 48, 65, 142, 165] [39, 49, 66, 143, 166] [40, 45, 67, 144, 167] [18, 60, 68, 121, 168] [19, 61, 69, 122, 169] [20, 62, 70, 123, 170] [63, 71, 92, 124, 171] [22, 65, 72, 151, 172] [23, 66, 73, 152, 173] [24, 67, 74, 153, 174] [51, 61, 75, 128, 175] [52, 62, 76, 129, 176] [53, 63, 77, 130, 177] [54, 59, 78, 131, 178] [56, 66, 79, 160, 179] [57, 67, 80, 161, 180] [1, 58, 81, 162, 181] [33, 76, 82, 137, 182] [34, 77, 83, 138, 183] [35, 78, 84, 139, 184] [38, 80, 85, 169, 185] [39, 81, 86, 170, 186] [69, 77, 87, 146, 187] [70, 78, 88, 147, 188] [5, 71, 89, 148, 189] [73, 81, 90, 176, 190] [6, 74, 91, 177, 191] [52, 88, 92, 156, 192] [53, 89, 93, 157, 193] [57, 91, 94, 183, 194] [83, 89, 95, 165, 195] [13, 84, 96, 166, 196] [15, 86, 97, 188, 197] [70, 96, 98, 173, 198] [25, 93, 99, 180, 199]
H_Z (100 checks, sparse supports)
[0, 19, 100, 106, 122] [1, 26, 101, 103, 181] [2, 11, 102, 113, 133] [3, 33, 103, 115, 137] [4, 34, 104, 116, 138] [5, 16, 105, 108, 189] [6, 42, 106, 110, 191] [7, 43, 107, 111, 115] [8, 22, 108, 125, 151] [9, 23, 109, 126, 152] [10, 51, 110, 128, 155] [11, 52, 111, 129, 156] [12, 53, 112, 130, 157] [13, 29, 113, 118, 196] [14, 30, 114, 119, 125] [15, 60, 115, 121, 197] [16, 61, 116, 122, 128] [17, 62, 117, 123, 129] [18, 37, 118, 141, 168] [19, 38, 119, 142, 169] [20, 39, 120, 143, 170] [21, 68, 104, 121, 145] [22, 69, 122, 146, 172] [23, 70, 123, 147, 173] [24, 71, 124, 148, 174] [25, 46, 125, 132, 199] [26, 47, 126, 133, 141] [27, 48, 127, 134, 142] [28, 75, 128, 136, 149] [29, 76, 129, 137, 145] [30, 77, 130, 138, 146] [31, 78, 131, 139, 147] [32, 55, 109, 132, 159] [33, 56, 133, 160, 182] [34, 57, 134, 161, 183] [35, 58, 135, 162, 184] [9, 36, 101, 111, 136] [37, 82, 112, 137, 164] [38, 83, 138, 165, 185] [39, 84, 139, 166, 186] [8, 40, 110, 140, 167] [41, 64, 141, 150, 163] [42, 65, 142, 151, 159] [43, 66, 143, 152, 160] [44, 67, 144, 153, 161] [14, 45, 100, 145, 167] [46, 87, 101, 146, 155] [47, 88, 147, 156, 164] [48, 89, 148, 157, 165] [13, 49, 149, 158, 166] [4, 50, 105, 119, 150] [51, 72, 120, 151, 175] [52, 73, 152, 176, 192] [53, 74, 153, 177, 193] [3, 54, 118, 154, 178] [20, 55, 107, 123, 155] [56, 92, 124, 156, 179] [57, 93, 157, 180, 194] [18, 58, 121, 158, 181] [7, 59, 102, 159, 178] [60, 79, 105, 160, 168] [61, 80, 161, 169, 175] [62, 81, 162, 170, 176] [6, 63, 163, 171, 177] [27, 64, 104, 106, 164] [65, 95, 107, 165, 172] [66, 96, 166, 173, 179] [25, 67, 167, 174, 180] [12, 68, 114, 134, 168] [69, 85, 135, 169, 187] [70, 86, 170, 188, 198] [10, 71, 132, 171, 189] [35, 72, 117, 139, 172] [73, 98, 140, 173, 190] [32, 74, 136, 174, 191] [17, 75, 109, 113, 175] [76, 90, 114, 176, 182] [77, 91, 177, 183, 187] [15, 78, 178, 184, 188] [44, 79, 112, 116, 179] [80, 99, 117, 180, 185] [41, 81, 181, 186, 190] [24, 82, 127, 153, 182] [83, 94, 154, 183, 195] [21, 84, 150, 184, 196] [54, 85, 131, 158, 185] [50, 86, 100, 186, 197] [31, 87, 120, 126, 187] [88, 97, 127, 188, 192] [28, 89, 189, 193, 195] [63, 90, 124, 130, 190] [59, 91, 131, 191, 194] [40, 92, 144, 171, 192] [36, 93, 102, 193, 199] [2, 94, 103, 149, 194] [49, 95, 135, 143, 195] [45, 96, 144, 196, 198] [5, 97, 140, 148, 197] [0, 98, 108, 163, 198] [1, 99, 154, 162, 199]
Code ID 200-8-10 · download JSON · raw on GitHub