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[[216,76,3]] d =
n
216
k
76
d
3
kd²/n
3.167
w
8
X/Z
1
g
0.0027
r
5.831
layers
2
swaps
734

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)
[42, 70, 93]
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)
[112, 125, 148]
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)×216 (2,4)×864 (3,0)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 216 (2,4): 864 (2,6): 1296 (3,0): 72 (3,4): 3240 (3,6): 11232 (3,8): 10800
trapping sets H_Z (1,4)×216 (2,4)×864 (3,0)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 216 (2,4): 864 (2,6): 1296 (3,0): 72 (3,4): 3240 (3,6): 11232 (3,8): 10800
witness diameter X 1.4142 · Z 1.0 (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 = 5.831
X checkZ checkqubit site (111)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 734 nearest-neighbor SWAPs per round in total, at most 9 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 ((C3 x C3) : C3) : (C2 x C2) = SmallGroup(108, 17) (order 108), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 216; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 5, 11, 32}, b = {0, 5, 7, 24} 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 2bga-nonabelian (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

[[216,76,3]] — two-block group-algebra code over ((C3 x C3) : C3) : (C2 x C2)

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| = 216

Parameters are also spelled out in provenance.construction of codes/216-76-3.json.

Parity checks

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