[[204,24,15]] · QEC Challenge

[[204,24,15]] d ≤

204

kd²/n

26.471

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (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, 15 qubits)

[28, 39, 40, 41, 87, 93, 94, 106, 108, 135, 144, 153, 163, 167, 176]

d_Z 15 · witness weight 15 (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, 15 qubits)

[7, 20, 22, 51, 54, 69, 83, 90, 111, 117, 148, 149, 161, 165, 197]

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, 25, 70, 89}, 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

[[204,24,15]] — 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/204-24-15.json.

Parity checks

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

H_X (102 checks, sparse supports)

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

H_Z (102 checks, sparse supports)

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

Code ID 204-24-15 · download JSON · raw on GitHub