[[222,4,21]] · QEC Challenge

[[222,4,21]] d ≤

222

kd²/n

7.946

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 21, d_Z ≤ 21 · 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 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)

[7, 16, 18, 21, 22, 32, 37, 59, 69, 94, 107, 108, 111, 118, 132, 138, 148, 177, 189, 217, 221]

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)

[7, 10, 25, 28, 31, 37, 43, 46, 67, 70, 76, 82, 85, 97, 109, 145, 153, 177, 187, 194, 212]

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 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)×111 (2,4)×333 (3,3)×111 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 111 (1,4): 111 (2,4): 333 (2,5): 1332 (2,6): 666 (3,3): 111 (3,4): 222 (3,5): 999 (3,6): 11211 (3,7): 17427 (3,8): 5661 (3,9): 1998 (3,10): 444

trapping sets H_Z (1,3)×111 (2,4)×333 (3,3)×111 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 111 (1,4): 111 (2,4): 333 (2,5): 1332 (2,6): 666 (3,3): 111 (3,4): 222 (3,5): 999 (3,6): 11211 (3,7): 17427 (3,8): 5661 (3,9): 1998 (3,10): 444

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C37 : C3 = SmallGroup(111, 1) (order 111), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 222; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 2, 11, 71}, b = {0, 1, 21} 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 2bga-nonabelian (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

[[222,4,21]] — two-block group-algebra code over C37 : 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| = 222

Parameters are also spelled out in provenance.construction of codes/222-4-21.json.

Parity checks

X-checks 111 (max weight 7) · Z-checks 111 (max weight 7)

H_X (111 checks, sparse supports)

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

H_Z (111 checks, sparse supports)

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

Code ID 222-4-21 · download JSON · raw on GitHub