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[[234,24,18]] d ≤
n
234
k
24
d
18
kd²/n
33.231
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 18 · witness weight 18 (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, 18 qubits)
[5, 11, 16, 23, 42, 52, 63, 67, 102, 114, 139, 148, 154, 156, 169, 173, 187, 233]
d_Z 18 · witness weight 18 (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, 18 qubits)
[20, 22, 47, 53, 66, 96, 100, 114, 119, 129, 141, 160, 168, 180, 183, 205, 213, 233]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 9 · H_Z 9
qubit degrees H_X 4–5 (mean 4.5) · H_Z 4–5 (mean 4.5)
trapping sets H_X (1,4)×117 (2,5)×117 (3,6)×819 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 117 (1,5): 117 (2,5): 117 (2,6): 702 (2,7): 2106 (2,8): 1170 (3,6): 819 (3,7): 3276 (3,8): 8073 (3,9): 32760 (3,10): 44811 (3,11): 16965 (3,12): 3744 (3,13): 1170
trapping sets H_Z (1,4)×117 (2,5)×117 (3,6)×819 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 117 (1,5): 117 (2,5): 117 (2,6): 702 (2,7): 2106 (2,8): 1170 (3,6): 819 (3,7): 3276 (3,8): 8073 (3,9): 32760 (3,10): 44811 (3,11): 16965 (3,12): 3744 (3,13): 1170

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C117 = SmallGroup(117, 2) (order 117), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 234; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 8, 28, 59, 94}, b = {0, 4, 58, 89} as element indices. Check weight 9. 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

[[234,24,18]] — two-block group-algebra code over C117

Direction & hypothesis

Cell weight-9plus 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| = 234

Parameters are also spelled out in provenance.construction of codes/234-24-18.json.

Parity checks

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