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[[196,16,17]] d ≤
n
196
k
16
d
17
kd²/n
23.592
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 17 · witness weight 17 (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, 17 qubits)
[15, 17, 30, 37, 53, 54, 81, 89, 93, 101, 112, 118, 130, 142, 154, 159, 173]
d_Z 17 · witness weight 17 (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, 17 qubits)
[0, 21, 35, 47, 55, 71, 88, 93, 105, 109, 125, 140, 154, 168, 174, 175, 193]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×196 (2,4)×98 (3,4)×196 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 196 (2,4): 98 (2,6): 2548 (3,4): 196 (3,6): 3822 (3,8): 46158 (3,10): 4312
trapping sets H_Z (1,4)×196 (2,4)×98 (3,4)×196 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 196 (2,4): 98 (2,6): 2548 (3,4): 196 (3,6): 3822 (3,8): 46158 (3,10): 4312

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C7 x D14 = SmallGroup(98, 3) (order 98), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 196; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 21, 81}, b = {0, 4, 28, 43} as element indices. Check weight 8. 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 ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[196,16,17]] — two-block group-algebra code over C7 x D14

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| = 196

Parameters are also spelled out in provenance.construction of codes/196-16-17.json.

Parity checks

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