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[[240,16,14]] d ≤
n
240
k
16
d
14
kd²/n
13.067
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 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)
[8, 23, 67, 68, 69, 84, 88, 100, 109, 128, 151, 191, 203, 238]
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)
[27, 60, 96, 108, 131, 132, 152, 172, 173, 177, 192, 216, 217, 221]
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)×240 (2,4)×1800 (3,3)×120 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 240 (2,4): 1800 (3,3): 120 (3,5): 17640 (3,7): 2400
trapping sets H_Z (1,3)×240 (2,4)×1800 (3,3)×120 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 240 (2,4): 1800 (3,3): 120 (3,5): 17640 (3,7): 2400

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C3 x ((C10 x C2) : C2) = SmallGroup(120, 20) (order 120), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 240; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 7, 12}, b = {0, 16, 96} 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 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

[[240,16,14]] — two-block group-algebra code over C3 x ((C10 x C2) : C2)

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| = 240

Parameters are also spelled out in provenance.construction of codes/240-16-14.json.

Parity checks

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