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[[270,94,3]] d =
n
270
k
94
d
3
kd²/n
3.133
w
8
X/Z
1
g
0.002
r
6.3246
layers
2
swaps
1088

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (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, 3 qubits)
[139, 146, 160]
d_Z 3 · witness weight 3 (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, 3 qubits)
[1, 6, 18]
certificate exact, d = 3 · CryptoMiniSat 5.14.7 SAT
X: no logical < 3 exists; Z: no logical < 3 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)×270 (2,4)×1080 (3,0)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 270 (2,4): 1080 (2,6): 1620 (3,0): 90 (3,4): 4050 (3,6): 14040 (3,8): 13500
trapping sets H_Z (1,4)×270 (2,4)×1080 (3,0)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 270 (2,4): 1080 (2,6): 1620 (3,0): 90 (3,4): 4050 (3,6): 14040 (3,8): 13500
witness diameter X 2.2361 · Z 1.4142 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
layout contributed by @MathysRennela · Simulated annealing over integer-grid site assignments with the repo's own research/local2d/fold_layout.search_layout, single layer at radius 4 then two layers at 7, plus a divisor scan of the two-block index convention Z_l x Z_m which gives the tighter embedding here; radius and per-site occupancy re-measured with the verifier's own rule. · 2026-10-13
r = 6.325
X checkZ checkqubit site (142)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 1088 nearest-neighbor SWAPs per round in total, at most 10 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C15 x C3 x C3 = SmallGroup(135, 5) (order 135), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 270; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 69, 92}, b = {0, 2, 22, 41} 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 2D-local bilayer (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

[[270,94,3]] — two-block group-algebra code over C15 x C3 x C3

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| = 270

Parameters are also spelled out in provenance.construction of codes/270-94-3.json.

Parity checks

X-checks 135 (max weight 8) · Z-checks 135 (max weight 8)
H_X (135 checks, sparse supports)
[0, 9, 87, 121, 135, 144, 260, 269] [1, 18, 53, 98, 136, 153, 237, 264] [0, 2, 87, 107, 135, 137, 260, 267] [3, 24, 109, 130, 138, 159, 223, 257] [4, 25, 110, 131, 139, 160, 170, 215] [5, 32, 70, 112, 140, 167, 251, 268] [1, 6, 53, 76, 136, 141, 237, 254] [7, 38, 78, 117, 142, 173, 189, 234] [8, 39, 79, 118, 143, 147, 174, 181] [2, 9, 107, 121, 137, 144, 267, 269] [3, 10, 109, 123, 138, 145, 223, 243] [4, 11, 110, 124, 139, 146, 170, 192] [12, 46, 62, 105, 147, 181, 245, 266] [13, 47, 125, 134, 140, 148, 167, 182] [14, 16, 48, 55, 149, 183, 195, 238] [5, 15, 70, 93, 140, 150, 251, 263] [16, 55, 95, 126, 151, 190, 206, 248] [17, 56, 96, 127, 152, 156, 191, 198] [6, 18, 76, 98, 141, 153, 254, 264] [7, 19, 78, 100, 142, 154, 189, 212] [8, 20, 79, 101, 143, 147, 155, 161] [21, 31, 63, 74, 156, 198, 214, 253] [22, 64, 102, 129, 135, 144, 157, 199] [3, 23, 24, 65, 158, 164, 200, 207] [10, 24, 123, 130, 145, 159, 243, 257] [11, 25, 124, 131, 146, 160, 192, 215] [12, 26, 62, 85, 147, 161, 245, 259] [13, 27, 125, 132, 140, 148, 150, 162] [14, 16, 28, 33, 149, 163, 195, 218] [29, 72, 88, 122, 151, 164, 190, 207] [30, 35, 73, 80, 152, 165, 191, 208] [31, 36, 74, 81, 166, 209, 221, 255] [15, 32, 93, 112, 150, 167, 263, 268] [16, 33, 95, 114, 151, 168, 206, 229] [17, 34, 96, 115, 152, 156, 169, 175] [35, 45, 80, 91, 170, 215, 231, 262] [36, 81, 116, 133, 136, 153, 171, 216] [7, 37, 38, 82, 172, 178, 217, 224] [19, 38, 100, 117, 154, 173, 212, 234] [20, 39, 101, 118, 155, 161, 174, 181] [21, 31, 40, 51, 156, 175, 214, 236] [22, 41, 102, 119, 135, 137, 157, 176] [3, 10, 23, 42, 158, 164, 177, 184] [43, 54, 89, 99, 138, 159, 178, 224] [12, 44, 46, 90, 139, 160, 179, 225] [13, 45, 47, 91, 180, 187, 226, 232] [26, 46, 85, 105, 161, 181, 259, 266] [27, 47, 132, 134, 150, 162, 167, 182] [28, 33, 48, 55, 163, 183, 218, 238] [29, 49, 88, 108, 151, 164, 168, 184] [30, 35, 50, 57, 152, 165, 169, 185] [31, 36, 51, 58, 166, 186, 221, 241] [5, 32, 52, 97, 171, 187, 216, 232] [53, 60, 98, 103, 172, 188, 217, 233] [54, 61, 99, 104, 189, 234, 244, 265] [33, 55, 114, 126, 168, 190, 229, 248] [34, 56, 115, 127, 169, 175, 191, 198] [35, 45, 57, 68, 170, 192, 231, 250] [36, 58, 116, 128, 136, 141, 171, 193] [7, 19, 37, 59, 172, 178, 194, 201] [60, 71, 103, 113, 142, 173, 195, 238] [21, 61, 63, 104, 143, 174, 196, 239] [22, 62, 64, 105, 197, 204, 240, 246] [40, 51, 63, 74, 175, 198, 236, 253] [41, 64, 119, 129, 137, 144, 176, 199] [10, 24, 42, 65, 177, 184, 200, 207] [43, 54, 66, 77, 138, 145, 178, 201] [12, 26, 44, 67, 139, 146, 179, 202] [13, 27, 45, 68, 180, 187, 203, 210] [0, 9, 69, 111, 148, 182, 204, 246] [29, 70, 72, 112, 149, 183, 205, 247] [30, 71, 73, 113, 206, 213, 248, 252] [49, 72, 108, 122, 168, 184, 190, 207] [50, 57, 73, 80, 169, 185, 191, 208] [51, 58, 74, 81, 186, 209, 241, 255] [5, 15, 52, 75, 171, 187, 193, 210] [53, 60, 76, 83, 172, 188, 194, 211] [54, 61, 77, 84, 189, 212, 244, 258] [17, 56, 78, 117, 196, 213, 239, 252] [79, 86, 118, 120, 197, 214, 240, 253] [57, 68, 80, 91, 192, 215, 250, 262] [58, 81, 128, 133, 141, 153, 193, 216] [19, 38, 59, 82, 194, 201, 217, 224] [60, 71, 83, 94, 142, 154, 195, 218] [21, 40, 61, 84, 143, 155, 196, 219] [22, 41, 62, 85, 197, 204, 220, 227] [1, 18, 86, 120, 157, 199, 221, 255] [43, 87, 89, 121, 158, 200, 222, 256] [44, 88, 90, 122, 223, 230, 257, 261] [66, 77, 89, 99, 145, 159, 201, 224] [26, 46, 67, 90, 146, 160, 202, 225] [27, 47, 68, 91, 203, 210, 226, 232] [0, 2, 69, 92, 148, 162, 204, 227] [29, 49, 70, 93, 149, 163, 205, 228] [30, 50, 71, 94, 206, 213, 229, 235] [4, 25, 95, 126, 165, 208, 230, 261] [52, 96, 97, 127, 166, 209, 231, 262] [15, 32, 75, 97, 193, 210, 216, 232] [76, 83, 98, 103, 194, 211, 217, 233] [77, 84, 99, 104, 212, 234, 258, 265] [17, 34, 78, 100, 196, 213, 219, 235] [79, 86, 101, 106, 197, 214, 220, 236] [37, 82, 102, 129, 222, 237, 256, 264] [83, 94, 103, 113, 154, 173, 218, 238] [40, 63, 84, 104, 155, 174, 219, 239] [41, 64, 85, 105, 220, 227, 240, 246] [1, 6, 86, 106, 157, 176, 221, 241] [43, 66, 87, 107, 158, 177, 222, 242] [44, 67, 88, 108, 223, 230, 243, 249] [8, 39, 109, 130, 179, 225, 244, 265] [69, 110, 111, 131, 180, 226, 245, 266] [2, 9, 92, 111, 162, 182, 227, 246] [49, 72, 93, 112, 163, 183, 228, 247] [50, 73, 94, 113, 229, 235, 248, 252] [4, 11, 95, 114, 165, 185, 230, 249] [52, 75, 96, 115, 166, 186, 231, 250] [14, 48, 116, 133, 188, 233, 251, 268] [34, 56, 100, 117, 219, 235, 239, 252] [101, 106, 118, 120, 220, 236, 240, 253] [37, 59, 102, 119, 222, 237, 242, 254] [6, 18, 106, 120, 176, 199, 241, 255] [66, 89, 107, 121, 177, 200, 242, 256] [67, 90, 108, 122, 243, 249, 257, 261] [8, 20, 109, 123, 179, 202, 244, 258] [69, 92, 110, 124, 180, 203, 245, 259] [23, 65, 125, 134, 205, 247, 260, 269] [11, 25, 114, 126, 185, 208, 249, 261] [75, 97, 115, 127, 186, 209, 250, 262] [14, 28, 116, 128, 188, 211, 251, 263] [59, 82, 119, 129, 242, 254, 256, 264] [20, 39, 123, 130, 202, 225, 258, 265] [92, 111, 124, 131, 203, 226, 259, 266] [23, 42, 125, 132, 205, 228, 260, 267] [28, 48, 128, 133, 211, 233, 263, 268] [42, 65, 132, 134, 228, 247, 267, 269]
H_Z (135 checks, sparse supports)
[0, 2, 22, 41, 135, 137, 204, 227] [1, 6, 36, 58, 136, 141, 221, 241] [2, 9, 41, 64, 137, 144, 227, 246] [3, 10, 43, 66, 138, 145, 158, 177] [4, 11, 44, 67, 139, 146, 230, 249] [5, 13, 15, 27, 140, 150, 187, 210] [6, 18, 58, 81, 141, 153, 241, 255] [7, 19, 60, 83, 142, 154, 172, 194] [8, 20, 61, 84, 143, 155, 244, 258] [0, 9, 22, 64, 135, 144, 204, 246] [10, 24, 66, 89, 145, 159, 177, 200] [11, 25, 67, 90, 146, 160, 249, 261] [8, 12, 20, 26, 147, 161, 179, 202] [13, 27, 69, 92, 148, 162, 180, 203] [14, 28, 70, 93, 149, 163, 251, 263] [15, 27, 32, 47, 150, 167, 210, 232] [16, 29, 33, 49, 149, 151, 163, 168] [17, 30, 34, 50, 152, 169, 213, 235] [1, 18, 36, 81, 136, 153, 221, 255] [19, 38, 83, 103, 154, 173, 194, 217] [20, 39, 84, 104, 155, 174, 258, 265] [17, 21, 34, 40, 156, 175, 196, 219] [22, 41, 86, 106, 157, 176, 197, 220] [23, 42, 87, 107, 158, 177, 260, 267] [3, 24, 43, 89, 138, 158, 159, 200] [4, 25, 44, 90, 139, 160, 230, 261] [20, 26, 39, 46, 161, 181, 202, 225] [27, 47, 92, 111, 162, 182, 203, 226] [28, 48, 93, 112, 163, 183, 263, 268] [23, 29, 42, 49, 164, 184, 205, 228] [30, 50, 95, 114, 165, 185, 206, 229] [31, 51, 96, 115, 156, 166, 175, 186] [5, 13, 32, 47, 140, 167, 187, 232] [33, 49, 55, 72, 163, 168, 183, 190] [34, 50, 56, 73, 169, 191, 235, 252] [4, 11, 35, 57, 165, 170, 185, 192] [36, 52, 58, 75, 166, 171, 186, 193] [37, 53, 59, 76, 172, 194, 237, 254] [7, 38, 60, 103, 142, 172, 173, 217] [8, 39, 61, 104, 143, 174, 244, 265] [34, 40, 56, 63, 175, 198, 219, 239] [41, 64, 106, 120, 176, 199, 220, 240] [42, 65, 107, 121, 177, 200, 267, 269] [37, 43, 59, 66, 178, 201, 222, 242] [44, 67, 109, 123, 179, 202, 223, 243] [45, 68, 110, 124, 170, 180, 192, 203] [8, 12, 39, 46, 147, 179, 181, 225] [13, 47, 69, 111, 148, 180, 182, 226] [14, 48, 70, 112, 149, 183, 251, 268] [42, 49, 65, 72, 184, 207, 228, 247] [50, 73, 114, 126, 185, 208, 229, 248] [51, 74, 115, 127, 175, 186, 198, 209] [45, 52, 68, 75, 187, 210, 231, 250] [53, 76, 116, 128, 136, 141, 188, 211] [7, 19, 54, 77, 178, 189, 201, 212] [16, 29, 55, 72, 149, 151, 183, 190] [17, 30, 56, 73, 152, 191, 213, 252] [11, 25, 57, 80, 185, 192, 208, 215] [58, 75, 81, 97, 186, 193, 209, 216] [59, 76, 82, 98, 194, 217, 254, 264] [14, 28, 60, 83, 188, 195, 211, 218] [61, 78, 84, 100, 189, 196, 212, 219] [62, 79, 85, 101, 147, 161, 197, 220] [17, 21, 56, 63, 156, 196, 198, 239] [22, 64, 86, 120, 157, 197, 199, 240] [23, 65, 87, 121, 158, 200, 260, 269] [59, 66, 82, 89, 201, 224, 242, 256] [67, 90, 123, 130, 202, 225, 243, 257] [68, 91, 124, 131, 192, 203, 215, 226] [62, 69, 85, 92, 204, 227, 245, 259] [70, 93, 125, 132, 140, 150, 205, 228] [16, 33, 71, 94, 195, 206, 218, 229] [23, 29, 65, 72, 164, 205, 207, 247] [30, 73, 95, 126, 165, 206, 208, 248] [31, 74, 96, 127, 156, 166, 198, 209] [68, 75, 91, 97, 210, 232, 250, 262] [76, 98, 128, 133, 141, 153, 211, 233] [19, 38, 77, 99, 201, 212, 224, 234] [71, 78, 94, 100, 142, 154, 213, 235] [21, 40, 79, 101, 143, 155, 214, 236] [4, 25, 35, 80, 165, 170, 208, 215] [36, 52, 81, 97, 166, 171, 209, 216] [37, 53, 82, 98, 172, 217, 237, 264] [28, 48, 83, 103, 211, 218, 233, 238] [84, 100, 104, 117, 212, 219, 234, 239] [85, 101, 105, 118, 161, 181, 220, 240] [31, 51, 86, 106, 214, 221, 236, 241] [87, 102, 107, 119, 135, 137, 222, 242] [3, 10, 88, 108, 164, 184, 223, 243] [37, 43, 82, 89, 178, 222, 224, 256] [44, 90, 109, 130, 179, 223, 225, 257] [45, 91, 110, 131, 170, 180, 215, 226] [85, 92, 105, 111, 227, 246, 259, 266] [93, 112, 132, 134, 150, 167, 228, 247] [33, 55, 94, 113, 218, 229, 238, 248] [88, 95, 108, 114, 151, 168, 230, 249] [35, 57, 96, 115, 152, 169, 231, 250] [45, 52, 91, 97, 187, 231, 232, 262] [53, 98, 116, 133, 136, 153, 188, 233] [7, 38, 54, 99, 178, 189, 224, 234] [94, 100, 113, 117, 154, 173, 235, 252] [40, 63, 101, 118, 155, 174, 236, 253] [1, 6, 102, 119, 157, 176, 237, 254] [14, 48, 60, 103, 188, 195, 233, 238] [61, 78, 104, 117, 189, 196, 234, 239] [62, 79, 105, 118, 147, 181, 197, 240] [51, 74, 106, 120, 236, 241, 253, 255] [107, 119, 121, 129, 137, 144, 242, 256] [10, 24, 108, 122, 184, 207, 243, 257] [54, 77, 109, 123, 138, 145, 244, 258] [12, 26, 110, 124, 139, 146, 245, 259] [62, 69, 105, 111, 204, 245, 246, 266] [70, 112, 125, 134, 140, 167, 205, 247] [16, 55, 71, 113, 195, 206, 238, 248] [108, 114, 122, 126, 168, 190, 249, 261] [57, 80, 115, 127, 169, 191, 250, 262] [5, 15, 116, 128, 171, 193, 251, 263] [71, 78, 113, 117, 142, 173, 213, 252] [21, 63, 79, 118, 143, 174, 214, 253] [6, 18, 119, 129, 176, 199, 254, 264] [31, 74, 86, 120, 214, 221, 253, 255] [87, 102, 121, 129, 135, 144, 222, 256] [3, 24, 88, 122, 164, 207, 223, 257] [77, 99, 123, 130, 145, 159, 258, 265] [26, 46, 124, 131, 146, 160, 259, 266] [0, 2, 125, 132, 148, 162, 260, 267] [88, 95, 122, 126, 151, 190, 230, 261] [35, 80, 96, 127, 152, 191, 231, 262] [15, 32, 128, 133, 193, 216, 263, 268] [1, 18, 102, 129, 157, 199, 237, 264] [54, 99, 109, 130, 138, 159, 244, 265] [12, 46, 110, 131, 139, 160, 245, 266] [2, 9, 132, 134, 162, 182, 267, 269] [5, 32, 116, 133, 171, 216, 251, 268] [0, 9, 125, 134, 148, 182, 260, 269]
Code ID 270-94-3 · download JSON · raw on GitHub