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[[252,168,2]] d =
n
252
k
168
d
2
kd²/n
2.667
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (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, 2 qubits)
[43, 62]
d_Z 2 · witness weight 2 (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, 2 qubits)
[106, 124]
certificate exact, d = 2 · CryptoMiniSat 5.14.7 SAT
X: no logical < 2 exists; Z: no logical < 2 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 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 3–6 (mean 4.5) · H_Z 3–6 (mean 4.5)
trapping sets H_X (1,3)×126 (2,0)×252 (3,0)×1134 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 126 (1,6): 126 (2,0): 252 (2,3): 756 (2,6): 378 (3,0): 1134 (3,3): 3066 (3,6): 2688 (3,9): 1134
trapping sets H_Z (1,3)×126 (2,0)×252 (3,0)×1134 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 126 (1,6): 126 (2,0): 252 (2,3): 756 (2,6): 378 (3,0): 1134 (3,3): 3066 (3,6): 2688 (3,9): 1134

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C126 = SmallGroup(126, 6) (order 126), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 252; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 4, 13, 45, 64, 83}, b = {0, 4, 13} 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

[[252,168,2]] — two-block group-algebra code over C126

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| = 252

Parameters are also spelled out in provenance.construction of codes/252-168-2.json.

Parity checks

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