[[234,8,21]] · QEC Challenge

[[234,8,21]] d ≤

234

kd²/n

15.077

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, 14, 18, 20, 36, 80, 88, 89, 91, 92, 105, 110, 117, 120, 121, 125, 137, 142, 209, 212, 224]

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)

[1, 4, 5, 7, 9, 24, 31, 59, 60, 131, 142, 146, 147, 150, 160, 171, 175, 179, 204, 221, 228]

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)×117 (2,4)×351 (3,3)×117 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 117 (1,4): 117 (2,4): 351 (2,5): 1404 (2,6): 702 (3,3): 117 (3,4): 117 (3,5): 1053 (3,6): 12012 (3,7): 18369 (3,8): 6435 (3,9): 2106 (3,10): 468

trapping sets H_Z (1,3)×117 (2,4)×351 (3,3)×117 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 117 (1,4): 117 (2,4): 351 (2,5): 1404 (2,6): 702 (3,3): 117 (3,4): 117 (3,5): 1053 (3,6): 12012 (3,7): 18369 (3,8): 6435 (3,9): 2106 (3,10): 468

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C3 x (C13 : C3) = SmallGroup(117, 3) (order 117), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 234; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 16, 19}, b = {0, 15, 88} 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

[[234,8,21]] — two-block group-algebra code over C3 x (C13 : 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| = 234

Parameters are also spelled out in provenance.construction of codes/234-8-21.json.

Parity checks

X-checks 117 (max weight 7) · Z-checks 117 (max weight 7)

H_X (117 checks, sparse supports)

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

H_Z (117 checks, sparse supports)

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

Code ID 234-8-21 · download JSON · raw on GitHub