[[270,36,13]] · QEC Challenge

[[270,36,13]] d ≤

270

kd²/n

22.533

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · 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 13 · witness weight 13 (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, 13 qubits)

[14, 15, 35, 50, 56, 81, 87, 104, 110, 123, 129, 132, 149]

d_Z 13 · witness weight 13 (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, 13 qubits)

[17, 144, 152, 163, 164, 165, 182, 189, 193, 209, 236, 254, 266]

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 9 · H_Z 9

qubit degrees H_X 3–6 (mean 4.5) · H_Z 3–6 (mean 4.5)

trapping sets H_X (1,3)×135 (2,4)×405 (3,3)×135 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 135 (1,6): 135 (2,4): 405 (2,7): 2430 (2,10): 2025 (3,3): 135 (3,5): 1215 (3,6): 405 (3,7): 135 (3,8): 26730 (3,9): 1080 (3,10): 2430 (3,11): 72090 (3,12): 4725 (3,13): 6075 (3,14): 36450 (3,16): 2700

trapping sets H_Z (1,3)×135 (2,4)×405 (3,3)×135 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C45 x C3 = SmallGroup(135, 2) (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, 6, 44, 46, 51, 111}, b = {0, 10, 98} 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

[[270,36,13]] — two-block group-algebra code over C45 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| = 270

Parameters are also spelled out in provenance.construction of codes/270-36-13.json.

Parity checks

X-checks 135 (max weight 9) · Z-checks 135 (max weight 9)

H_X (135 checks, sparse supports)

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

H_Z (135 checks, sparse supports)

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

Code ID 270-36-13 · download JSON · raw on GitHub