[[196,12,20]] · QEC Challenge

[[196,12,20]] d ≤

196

kd²/n

24.49

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 20, d_Z ≤ 20 · 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 20 · witness weight 20 (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, 20 qubits)

[10, 12, 44, 47, 48, 52, 65, 76, 79, 85, 100, 102, 110, 114, 129, 141, 169, 175, 181, 183]

d_Z 20 · witness weight 20 (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, 20 qubits)

[58, 59, 64, 79, 83, 103, 109, 111, 115, 125, 136, 150, 154, 157, 169, 174, 184, 188, 190, 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 8 · H_Z 8

qubit degrees H_X 3–5 (mean 4.0) · H_Z 3–5 (mean 4.0)

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 (1,5): 98 (2,4): 294 (2,6): 1519 (2,8): 882 (3,3): 98 (3,5): 1176 (3,7): 15680 (3,9): 29400 (3,11): 13818 (3,13): 686

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

(1,3): 98 (1,5): 98 (2,4): 294 (2,6): 1519 (2,8): 882 (3,3): 98 (3,5): 1176 (3,7): 15680 (3,9): 29400 (3,11): 13818 (3,13): 686

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, 1, 13, 32, 57}, b = {0, 3, 59} 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

[[196,12,20]] — 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/196-12-20.json.

Parity checks

X-checks 98 (max weight 8) · Z-checks 98 (max weight 8)

H_X (98 checks, sparse supports)

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

H_Z (98 checks, sparse supports)

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

Code ID 196-12-20 · download JSON · raw on GitHub