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[[270,24,12]] d ≤
n
270
k
24
d
12
kd²/n
12.8
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 12 · witness weight 12 (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, 12 qubits)
[1, 2, 3, 32, 33, 38, 231, 236, 237, 266, 267, 268]
d_Z 12 · witness weight 12 (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, 12 qubits)
[5, 6, 9, 12, 57, 63, 189, 243, 245, 248, 250, 253]
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 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×135 (2,4)×405 (3,3)×135 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 135 (1,4): 135 (2,4): 405 (2,5): 1620 (2,6): 810 (3,3): 135 (3,5): 1215 (3,6): 14355 (3,7): 21195 (3,8): 7155 (3,9): 2430 (3,10): 540
trapping sets H_Z (1,3)×135 (2,4)×405 (3,3)×135 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 135 (1,4): 135 (2,4): 405 (2,5): 1620 (2,6): 810 (3,3): 135 (3,5): 1215 (3,6): 14355 (3,7): 21195 (3,8): 7155 (3,9): 2430 (3,10): 540

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, 13, 122, 125}, b = {0, 29, 59} as element indices. Check weight 7. 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

[[270,24,12]] — 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/270-24-12.json.

Parity checks

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