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[[252,20,21]] d ≤
n
252
k
20
d
21
kd²/n
35.0
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 21, d_Z ≤ 21 · 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 21 · witness weight 21 (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, 21 qubits)
[1, 3, 8, 27, 49, 51, 88, 100, 110, 120, 122, 149, 155, 160, 165, 211, 219, 224, 231, 232, 244]
d_Z 21 · witness weight 21 (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, 21 qubits)
[12, 16, 20, 38, 60, 62, 68, 73, 139, 150, 156, 161, 162, 179, 190, 196, 201, 222, 242, 245, 251]
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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×252 (2,6)×3528 (3,6)×1260 (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 (2,6): 3528 (3,6): 1260 (3,8): 70308 (3,10): 7056
trapping sets H_Z (1,4)×252 (2,6)×3528 (3,6)×1260 (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 (2,6): 3528 (3,6): 1260 (3,8): 70308 (3,10): 7056

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, 18, 71, 101}, b = {0, 23, 52, 112} 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

[[252,20,21]] — 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/252-20-21.json.

Parity checks

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