[[204,28,12]] · QEC Challenge

[[204,28,12]] d ≤

204

kd²/n

19.765

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (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, 12 qubits)

[14, 47, 57, 58, 110, 116, 142, 149, 168, 180, 196, 201]

d_Z 12 · witness weight 12 (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, 12 qubits)

[1, 16, 30, 42, 53, 59, 79, 85, 121, 154, 155, 159]

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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×204 (2,6)×2856 (3,6)×2040 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 204 (2,6): 2856 (3,6): 2040 (3,8): 53856 (3,10): 5712

trapping sets H_Z (1,4)×204 (2,6)×2856 (3,6)×2040 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 204 (2,6): 2856 (3,6): 2040 (3,8): 53856 (3,10): 5712

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C102 = SmallGroup(102, 4) (order 102), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 204; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 28, 70, 86}, b = {0, 50, 61, 97} 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 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

[[204,28,12]] — two-block group-algebra code over C102

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| = 204

Parameters are also spelled out in provenance.construction of codes/204-28-12.json.

Parity checks

X-checks 102 (max weight 8) · Z-checks 102 (max weight 8)

H_X (102 checks, sparse supports)

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

H_Z (102 checks, sparse supports)

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

Code ID 204-28-12 · download JSON · raw on GitHub