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[[252,128,2]] d =
n
252
k
128
d
2
kd²/n
2.032
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 8, w_Z = 8 (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)
[40, 49]
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)
[44, 53]
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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×252 (2,0)×126 (3,4)×10584 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 252 (2,0): 126 (2,4): 1512 (3,4): 10584 (3,8): 2016
trapping sets H_Z (1,4)×252 (2,0)×126 (3,4)×10584 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 252 (2,0): 126 (2,4): 1512 (3,4): 10584 (3,8): 2016

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, 1, 54, 63}, b = {0, 1, 36, 45} as element indices. Check weight 8. 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

[[252,128,2]] — two-block group-algebra code over C126

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| = 252

Parameters are also spelled out in provenance.construction of codes/252-128-2.json.

Parity checks

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