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[[294,12,7]] d =
n
294
k
12
d
7
kd²/n
2.0
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 7, d_Z = 7 · 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 7 · witness weight 7 (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, 7 qubits)
[45, 62, 81, 99, 114, 126, 135]
d_Z 7 · witness weight 7 (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, 7 qubits)
[151, 157, 166, 178, 193, 211, 230]
certificate exact, d = 7 · CryptoMiniSat 5.14 SAT
X: no logical < 7 exists; Z: no logical < 7 exists

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 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)×147 (2,2)×147 (3,2)×147 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 147 (1,3): 147 (2,2): 147 (2,3): 882 (2,4): 441 (3,2): 147 (3,3): 2695 (3,4): 4851 (3,5): 2058 (3,6): 882 (3,7): 147
trapping sets H_Z (1,2)×147 (2,2)×147 (3,2)×147 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 147 (1,3): 147 (2,2): 147 (2,3): 882 (2,4): 441 (3,2): 147 (3,3): 2695 (3,4): 4851 (3,5): 2058 (3,6): 882 (3,7): 147

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C7 x (C7 : C3) = SmallGroup(147, 3) (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, 1, 8}, b = {0, 2} 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

[[294,12,7]] — two-block group-algebra code over C7 x (C7 : C3)

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/294-12-7.json.

Parity checks

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