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[[210,6,13]] d ≤
n
210
k
6
d
13
kd²/n
4.829
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · 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 13 · witness weight 13 (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, 13 qubits)
[8, 11, 16, 20, 36, 50, 72, 77, 98, 115, 125, 154, 194]
d_Z 13 · witness weight 13 (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, 13 qubits)
[25, 35, 50, 103, 138, 146, 170, 174, 194, 199, 202, 205, 208]
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)×105 (2,2)×105 (3,2)×105 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 105 (1,3): 105 (2,2): 105 (2,3): 630 (2,4): 315 (3,2): 105 (3,3): 1995 (3,4): 3465 (3,5): 1260 (3,6): 630 (3,7): 105
trapping sets H_Z (1,2)×105 (2,2)×105 (3,2)×105 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 105 (1,3): 105 (2,2): 105 (2,3): 630 (2,4): 315 (3,2): 105 (3,3): 1995 (3,4): 3465 (3,5): 1260 (3,6): 630 (3,7): 105

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C105 = SmallGroup(105, 2) (order 105), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 210; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 3, 29}, b = {0, 12} 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

[[210,6,13]] — two-block group-algebra code over C105

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| = 210

Parameters are also spelled out in provenance.construction of codes/210-6-13.json.

Parity checks

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