[[252,32,16]] · QEC Challenge

[[252,32,16]] d ≤

252

kd²/n

32.508

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 16 · witness weight 16 (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, 16 qubits)

[1, 5, 28, 45, 89, 122, 154, 156, 181, 183, 203, 205, 229, 231, 245, 246]

d_Z 16 · witness weight 16 (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, 16 qubits)

[18, 27, 44, 63, 65, 67, 75, 92, 93, 94, 131, 158, 170, 171, 207, 217]

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 9 · H_Z 9

qubit degrees H_X 4–5 (mean 4.5) · H_Z 4–5 (mean 4.5)

trapping sets H_X (1,4)×126 (2,5)×126 (3,5)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 126 (1,5): 126 (2,5): 126 (2,6): 756 (2,7): 2268 (2,8): 1260 (3,5): 252 (3,6): 798 (3,7): 2898 (3,8): 9072 (3,9): 35196 (3,10): 47376 (3,11): 18900 (3,12): 4032 (3,13): 1260

trapping sets H_Z (1,4)×126 (2,5)×126 (3,5)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 126 (1,5): 126 (2,5): 126 (2,6): 756 (2,7): 2268 (2,8): 1260 (3,5): 252 (3,6): 798 (3,7): 2898 (3,8): 9072 (3,9): 35196 (3,10): 47376 (3,11): 18900 (3,12): 4032 (3,13): 1260

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C42 x C3 = SmallGroup(126, 16) (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, 5, 22, 78, 117}, b = {0, 4, 102, 121} as element indices. Check weight 9. 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

[[252,32,16]] — two-block group-algebra code over C42 x C3

Direction & hypothesis

Cell weight-9plus 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/252-32-16.json.

Parity checks

X-checks 126 (max weight 9) · Z-checks 126 (max weight 9)

H_X (126 checks, sparse supports)

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

H_Z (126 checks, sparse supports)

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

Code ID 252-32-16 · download JSON · raw on GitHub