[[300,8,14]] · QEC Challenge

[[300,8,14]] d ≤

300

kd²/n

5.227

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 14 · witness weight 14 (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, 14 qubits)

[8, 11, 91, 94, 113, 116, 132, 146, 161, 181, 186, 214, 250, 296]

d_Z 14 · witness weight 14 (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, 14 qubits)

[21, 34, 51, 62, 103, 137, 150, 171, 190, 193, 280, 283, 287, 293]

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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 5 · H_Z 5

qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)

trapping sets H_X (1,2)×150 (2,2)×150 (3,2)×150 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 150 (1,3): 150 (2,2): 150 (2,3): 900 (2,4): 450 (3,2): 150 (3,3): 2700 (3,4): 4950 (3,5): 2250 (3,6): 900 (3,7): 150

trapping sets H_Z (1,2)×150 (2,2)×150 (3,2)×150 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 150 (1,3): 150 (2,2): 150 (2,3): 900 (2,4): 450 (3,2): 150 (3,3): 2700 (3,4): 4950 (3,5): 2250 (3,6): 900 (3,7): 150

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C5 x D30 = SmallGroup(150, 11) (order 150), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 300; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 5, 42}, b = {0, 44} as element indices. Check weight 5. 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 ≤ 6 (computed)

How this code was found

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

[[300,8,14]] — two-block group-algebra code over C5 x D30

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| = 300

Parameters are also spelled out in provenance.construction of codes/300-8-14.json.

Parity checks

X-checks 150 (max weight 5) · Z-checks 150 (max weight 5)

H_X (150 checks, sparse supports)

[0, 50, 144, 150, 293] [1, 38, 78, 151, 296] [1, 2, 148, 152, 224] [3, 75, 117, 153, 298] [4, 64, 76, 154, 212] [0, 5, 101, 155, 237] [6, 62, 103, 156, 299] [7, 63, 104, 157, 225] [5, 8, 99, 158, 248] [6, 9, 132, 159, 249] [7, 10, 88, 153, 160] [11, 100, 134, 161, 282] [12, 20, 101, 162, 238] [13, 91, 102, 163, 239] [2, 14, 18, 164, 260] [3, 15, 121, 165, 261] [4, 16, 122, 156, 166] [17, 52, 88, 167, 288] [18, 89, 124, 168, 250] [19, 90, 125, 169, 251] [14, 20, 120, 170, 269] [15, 21, 49, 171, 270] [11, 16, 22, 159, 172] [17, 23, 143, 173, 199] [18, 24, 38, 161, 174] [19, 25, 112, 162, 175] [26, 39, 121, 176, 188] [27, 40, 122, 177, 262] [28, 115, 123, 178, 263] [8, 29, 33, 179, 278] [9, 30, 35, 180, 279] [10, 31, 36, 165, 181] [11, 32, 76, 182, 211] [12, 33, 136, 167, 183] [13, 34, 137, 168, 184] [35, 79, 112, 185, 200] [36, 113, 139, 186, 271] [37, 114, 140, 187, 272] [29, 38, 135, 188, 284] [30, 39, 72, 189, 285] [23, 31, 40, 171, 190] [32, 41, 74, 191, 222] [0, 33, 42, 173, 192] [26, 34, 43, 174, 193] [35, 44, 62, 150, 194] [36, 45, 63, 176, 195] [37, 46, 130, 177, 196] [47, 65, 136, 197, 213] [48, 66, 137, 198, 280] [49, 133, 138, 199, 281] [20, 50, 54, 200, 290] [21, 51, 56, 201, 291] [22, 52, 57, 180, 202] [7, 23, 53, 203, 235] [24, 54, 59, 182, 204] [25, 55, 60, 183, 205] [26, 56, 102, 151, 206] [27, 57, 145, 185, 207] [28, 58, 146, 186, 208] [59, 105, 130, 209, 226] [60, 131, 147, 210, 286] [51, 61, 132, 211, 287] [51, 62, 96, 212, 294] [41, 52, 63, 189, 213] [53, 64, 98, 214, 246] [2, 54, 65, 191, 215] [44, 55, 66, 192, 216] [3, 56, 67, 152, 217] [4, 57, 68, 194, 218] [47, 58, 69, 195, 219] [59, 70, 89, 154, 220] [60, 71, 90, 197, 221] [61, 72, 142, 198, 222] [73, 92, 145, 223, 240] [74, 93, 146, 224, 292] [39, 75, 80, 225, 297] [40, 76, 81, 201, 226] [16, 41, 77, 227, 258] [42, 78, 83, 203, 228] [43, 79, 84, 204, 229] [19, 44, 80, 155, 230] [45, 81, 86, 206, 231] [46, 82, 87, 207, 232] [47, 83, 123, 157, 233] [48, 84, 149, 209, 234] [49, 75, 85, 210, 235] [86, 126, 142, 236, 252] [77, 87, 143, 237, 295] [77, 88, 119, 238, 267] [8, 78, 89, 214, 239] [67, 79, 90, 215, 240] [9, 80, 91, 158, 241] [10, 81, 92, 217, 242] [70, 82, 93, 218, 243] [12, 83, 94, 160, 244] [13, 84, 95, 220, 245] [73, 85, 96, 221, 246] [86, 97, 113, 163, 247] [87, 98, 114, 223, 248] [99, 116, 149, 249, 264] [31, 64, 100, 250, 276] [65, 101, 106, 227, 251] [66, 102, 107, 228, 252] [34, 67, 103, 164, 253] [68, 104, 109, 230, 254] [69, 105, 110, 231, 255] [37, 70, 106, 166, 256] [71, 107, 111, 233, 257] [6, 72, 108, 234, 258] [73, 109, 138, 169, 259] [74, 100, 110, 236, 260] [29, 111, 148, 261, 273] [21, 103, 112, 170, 262] [22, 104, 113, 241, 263] [94, 105, 114, 242, 264] [24, 106, 115, 172, 265] [25, 107, 116, 244, 266] [97, 108, 117, 245, 267] [27, 109, 118, 175, 268] [28, 110, 119, 247, 269] [111, 120, 131, 178, 270] [55, 91, 121, 179, 271] [92, 122, 127, 253, 272] [93, 123, 128, 254, 273] [58, 94, 124, 181, 274] [95, 125, 129, 256, 275] [15, 96, 126, 257, 276] [61, 97, 127, 184, 277] [17, 98, 128, 259, 278] [50, 99, 129, 187, 279] [42, 124, 130, 190, 280] [43, 125, 131, 265, 281] [118, 126, 132, 266, 282] [45, 127, 133, 193, 283] [46, 128, 134, 268, 284] [48, 129, 135, 196, 285] [82, 115, 136, 202, 286] [116, 137, 141, 274, 287] [30, 117, 138, 275, 288] [85, 118, 139, 205, 289] [32, 119, 140, 277, 290] [1, 120, 141, 208, 291] [68, 139, 142, 216, 292] [69, 140, 143, 283, 293] [71, 141, 144, 219, 294] [108, 133, 145, 229, 295] [53, 134, 146, 289, 296] [5, 135, 147, 232, 297] [95, 147, 148, 243, 298] [14, 144, 149, 255, 299]

H_Z (150 checks, sparse supports)

[0, 44, 150, 155, 192] [1, 56, 151, 152, 291] [2, 67, 152, 164, 215] [3, 10, 153, 165, 217] [4, 70, 154, 166, 218] [5, 80, 155, 158, 297] [6, 16, 156, 159, 258] [7, 83, 157, 160, 203] [8, 91, 158, 179, 239] [9, 22, 159, 180, 241] [10, 94, 160, 181, 242] [11, 24, 161, 172, 182] [12, 25, 162, 183, 244] [13, 97, 163, 184, 245] [14, 103, 164, 170, 299] [15, 31, 165, 171, 276] [16, 106, 166, 172, 227] [17, 33, 167, 173, 278] [18, 34, 164, 168, 174] [19, 109, 169, 175, 230] [20, 112, 162, 170, 200] [21, 40, 171, 201, 262] [22, 115, 172, 202, 263] [23, 42, 173, 190, 203] [24, 43, 174, 204, 265] [25, 118, 175, 205, 266] [26, 45, 176, 193, 206] [27, 46, 177, 207, 268] [28, 120, 178, 208, 269] [29, 121, 179, 188, 261] [30, 52, 180, 189, 288] [31, 124, 181, 190, 250] [32, 54, 182, 191, 290] [33, 55, 179, 183, 192] [34, 127, 184, 193, 253] [35, 57, 180, 185, 194] [36, 58, 181, 186, 195] [37, 129, 187, 196, 256] [26, 38, 151, 174, 188] [39, 63, 176, 189, 225] [40, 130, 177, 190, 226] [41, 65, 191, 213, 227] [42, 66, 192, 228, 280] [43, 133, 193, 229, 281] [44, 68, 194, 216, 230] [45, 69, 195, 231, 283] [46, 135, 196, 232, 284] [47, 71, 197, 219, 233] [48, 72, 198, 234, 285] [23, 49, 171, 199, 235] [35, 50, 150, 200, 279] [51, 76, 201, 211, 212] [52, 136, 167, 202, 213] [53, 78, 203, 214, 296] [54, 79, 200, 204, 215] [55, 139, 205, 216, 271] [56, 81, 201, 206, 217] [57, 82, 202, 207, 218] [58, 141, 208, 219, 274] [59, 84, 204, 209, 220] [60, 85, 205, 210, 221] [32, 61, 211, 222, 277] [4, 62, 156, 194, 212] [47, 63, 157, 195, 213] [64, 89, 154, 214, 250] [65, 90, 197, 215, 251] [66, 142, 198, 216, 252] [67, 92, 217, 240, 253] [68, 93, 218, 254, 292] [69, 144, 219, 255, 293] [70, 95, 220, 243, 256] [71, 96, 221, 257, 294] [41, 72, 189, 222, 258] [73, 98, 223, 246, 259] [2, 74, 191, 224, 260] [7, 75, 153, 225, 235] [59, 76, 154, 182, 226] [77, 101, 227, 237, 238] [78, 102, 151, 228, 239] [79, 145, 185, 229, 240] [80, 104, 225, 230, 241] [81, 105, 226, 231, 242] [82, 147, 232, 243, 286] [83, 107, 228, 233, 244] [84, 108, 229, 234, 245] [53, 85, 235, 246, 289] [86, 110, 231, 236, 247] [5, 87, 232, 237, 248] [12, 88, 160, 167, 238] [13, 89, 168, 220, 239] [73, 90, 169, 221, 240] [91, 113, 163, 241, 271] [92, 114, 223, 242, 272] [93, 148, 224, 243, 273] [94, 116, 244, 264, 274] [95, 117, 245, 275, 298] [64, 96, 212, 246, 276] [97, 119, 247, 267, 277] [8, 98, 214, 248, 278] [9, 99, 158, 249, 279] [18, 100, 161, 250, 260] [19, 101, 155, 162, 251] [86, 102, 163, 206, 252] [103, 122, 156, 253, 262] [104, 123, 157, 254, 263] [105, 149, 209, 255, 264] [106, 125, 251, 256, 265] [107, 126, 252, 257, 266] [77, 108, 258, 267, 295] [109, 128, 254, 259, 268] [14, 110, 255, 260, 269] [15, 111, 257, 261, 270] [27, 112, 175, 185, 262] [28, 113, 186, 247, 263] [99, 114, 187, 248, 264] [115, 131, 178, 265, 286] [116, 132, 249, 266, 287] [88, 117, 153, 267, 288] [118, 134, 268, 282, 289] [20, 119, 238, 269, 290] [21, 120, 170, 270, 291] [36, 121, 165, 176, 271] [37, 122, 166, 177, 272] [111, 123, 178, 233, 273] [124, 137, 168, 274, 280] [125, 138, 169, 275, 281] [100, 126, 236, 276, 282] [127, 140, 272, 277, 283] [29, 128, 273, 278, 284] [30, 129, 275, 279, 285] [48, 130, 196, 209, 280] [49, 131, 210, 270, 281] [11, 132, 159, 211, 282] [133, 143, 199, 283, 295] [38, 134, 161, 284, 296] [39, 135, 188, 285, 297] [60, 136, 183, 197, 286] [61, 137, 184, 198, 287] [17, 138, 199, 259, 288] [139, 146, 186, 289, 292] [50, 140, 187, 290, 293] [51, 141, 287, 291, 294] [74, 142, 222, 236, 292] [0, 143, 173, 237, 293] [62, 144, 150, 294, 299] [87, 145, 207, 223, 295] [1, 146, 208, 224, 296] [75, 147, 210, 297, 298] [3, 148, 152, 261, 298] [6, 149, 234, 249, 299]

Code ID 300-8-14 · download JSON · raw on GitHub