[[196,12,19]] · QEC Challenge

[[196,12,19]] d ≤

196

kd²/n

22.102

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 19 · witness weight 19 (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, 19 qubits)

[10, 11, 20, 23, 29, 30, 53, 69, 72, 78, 119, 138, 150, 166, 167, 175, 185, 188, 190]

d_Z 19 · witness weight 19 (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, 19 qubits)

[9, 10, 11, 16, 17, 22, 74, 91, 97, 113, 117, 129, 140, 147, 148, 149, 172, 184, 195]

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 9 · H_Z 9

qubit degrees H_X 3–6 (mean 4.5) · H_Z 3–6 (mean 4.5)

trapping sets H_X (1,3)×98 (2,4)×294 (3,3)×98 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,3): 98 (1,6): 98 (2,4): 294 (2,7): 1764 (2,8): 196 (2,10): 1078 (3,3): 98 (3,5): 882 (3,6): 294 (3,7): 294 (3,8): 19502 (3,9): 5194 (3,10): 2842 (3,11): 41258 (3,12): 6370 (3,13): 3234 (3,14): 13720 (3,16): 784

trapping sets H_Z (1,3)×98 (2,4)×294 (3,3)×98 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C14 x C7 = SmallGroup(98, 5) (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, 9, 41, 53, 56, 85}, b = {0, 19, 67} as element indices. Check weight 9. 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 > 8 (computed)

How this code was found

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

[[196,12,19]] — two-block group-algebra code over C14 x C7

Direction & hypothesis

Cell weight-9plus 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/196-12-19.json.

Parity checks

X-checks 98 (max weight 9) · Z-checks 98 (max weight 9)

H_X (98 checks, sparse supports)

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

H_Z (98 checks, sparse supports)

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

Code ID 196-12-19 · download JSON · raw on GitHub