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[[270,34,14]] d ≤
n
270
k
34
d
14
kd²/n
24.681
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 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)
[35, 55, 74, 77, 100, 102, 140, 155, 168, 178, 185, 211, 232, 252]
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)
[15, 41, 60, 62, 63, 98, 135, 173, 185, 197, 204, 233, 236, 251]
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)×270 (2,6)×3780 (3,6)×2115 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 270 (2,6): 3780 (3,6): 2115 (3,8): 73035 (3,10): 7560
trapping sets H_Z (1,4)×270 (2,6)×3780 (3,6)×2115 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 270 (2,6): 3780 (3,6): 2115 (3,8): 73035 (3,10): 7560

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C15 x C3 x C3 = SmallGroup(135, 5) (order 135), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 270; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 10, 23, 129}, b = {0, 25, 60, 115} 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

[[270,34,14]] — two-block group-algebra code over C15 x C3 x 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| = 270

Parameters are also spelled out in provenance.construction of codes/270-34-14.json.

Parity checks

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