[[216,76,2]] · QEC Challenge

[[216,76,2]] d =

216

kd²/n

1.407

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 7, w_Z = 7 (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)

[154, 182]

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)

[73, 94]

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 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×108 (2,0)×162 (3,3)×684 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,3): 108 (1,4): 108 (2,0): 162 (2,4): 216 (2,5): 1296 (3,3): 684 (3,4): 1944 (3,5): 2592 (3,6): 5832 (3,7): 6480

trapping sets H_Z (1,3)×108 (2,0)×162 (3,3)×684 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,3): 108 (1,4): 108 (2,0): 162 (2,4): 216 (2,5): 1296 (3,3): 684 (3,4): 1944 (3,5): 2592 (3,6): 5832 (3,7): 6480

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C36 x C3 = SmallGroup(108, 12) (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, 4, 67, 89}, b = {0, 16, 65} as element indices. Check weight 7. 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

[[216,76,2]] — two-block group-algebra code over C36 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| = 216

Parameters are also spelled out in provenance.construction of codes/216-76-2.json.

Parity checks

X-checks 108 (max weight 7) · Z-checks 108 (max weight 7)

H_X (108 checks, sparse supports)

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

H_Z (108 checks, sparse supports)

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

Code ID 216-76-2 · download JSON · raw on GitHub