[[294,6,24]] · QEC Challenge

[[294,6,24]] d ≤

294

kd²/n

11.755

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 24, d_Z ≤ 24 · 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 24 · witness weight 24 (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, 24 qubits)

[10, 12, 38, 42, 48, 52, 62, 88, 124, 128, 140, 165, 166, 169, 197, 201, 215, 218, 234, 239, 248, 251, 265, 281]

d_Z 24 · witness weight 24 (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, 24 qubits)

[6, 39, 60, 70, 81, 84, 99, 102, 114, 118, 126, 148, 154, 160, 164, 173, 213, 217, 223, 226, 234, 258, 271, 284]

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)×588 (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): 588 (3,5): 20286 (3,7): 2940

trapping sets H_Z (1,3)×294 (2,4)×2205 (3,3)×588 (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): 588 (3,5): 20286 (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, 2, 51}, b = {0, 32, 73} 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

[[294,6,24]] — 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/294-6-24.json.

Parity checks

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

H_X (147 checks, sparse supports)

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

H_Z (147 checks, sparse supports)

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

Code ID 294-6-24 · download JSON · raw on GitHub