[[294,10,20]] · QEC Challenge

[[294,10,20]] d ≤

294

kd²/n

13.605

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 20, d_Z ≤ 20 · w_X = 6, w_Z = 6 (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)

[2, 7, 11, 14, 25, 27, 32, 35, 39, 51, 62, 70, 82, 118, 125, 138, 144, 218, 230, 258]

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)

[16, 20, 35, 52, 64, 68, 81, 88, 109, 123, 129, 150, 160, 163, 192, 212, 222, 251, 269, 277]

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 6 · H_Z 6

qubit degrees H_X 3 · H_Z 3

trapping sets H_X (1,3)×294 (2,4)×2205 (3,3)×294 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 294 (2,4): 2205 (3,3): 294 (3,5): 21168 (3,7): 2940

trapping sets H_Z (1,3)×294 (2,4)×2205 (3,3)×294 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 294 (2,4): 2205 (3,3): 294 (3,5): 21168 (3,7): 2940

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C147 = SmallGroup(147, 2) (order 147), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 294; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 5, 31}, b = {0, 22, 49} as element indices. Check weight 6. 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 ≤ 6 (computed)

How this code was found

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

[[294,10,20]] — two-block group-algebra code over C147

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| = 294

Parameters are also spelled out in provenance.construction of codes/294-10-20.json.

Parity checks

X-checks 147 (max weight 6) · Z-checks 147 (max weight 6)

H_X (147 checks, sparse supports)

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

H_Z (147 checks, sparse supports)

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

Code ID 294-10-20 · download JSON · raw on GitHub