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[[240,10,7]] d ≤
n
240
k
10
d
7
kd²/n
2.042
w
5
X/Z
1.14

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Distance

X/Z asymmetry 1.14 · d_X ≤ 8, 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 8 · witness weight 8 (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, 8 qubits)
[39, 62, 73, 75, 94, 96, 115, 119]
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)
[71, 99, 155, 164, 191, 214, 235]
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 8 · H_Z 8 (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)×120 (2,2)×120 (3,2)×120 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 120 (1,3): 120 (2,2): 120 (2,3): 720 (2,4): 360 (3,2): 120 (3,3): 2160 (3,4): 3960 (3,5): 1800 (3,6): 720 (3,7): 120
trapping sets H_Z (1,2)×120 (2,2)×120 (3,2)×120 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 120 (1,3): 120 (2,2): 120 (2,3): 720 (2,4): 360 (3,2): 120 (3,3): 2160 (3,4): 3960 (3,5): 1800 (3,6): 720 (3,7): 120

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C3 x (C5 : C8) = SmallGroup(120, 6) (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, 6, 28}, b = {0, 7} 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

[[240,10,7]] — two-block group-algebra code over C3 x (C5 : C8)

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/240-10-7.json.

Parity checks

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