[[248,20,13]] · QEC Challenge

[[248,20,13]] d ≤

248

kd²/n

13.629

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · 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 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)

[18, 48, 62, 71, 98, 110, 143, 147, 165, 187, 194, 195, 223]

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)

[1, 25, 77, 132, 145, 146, 156, 174, 184, 217, 236, 241, 246]

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 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)×124 (2,3)×124 (3,3)×124 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 124 (1,4): 124 (2,3): 124 (2,4): 372 (2,5): 1240 (2,6): 744 (3,3): 124 (3,4): 1116 (3,5): 2852 (3,6): 10168 (3,7): 14632 (3,8): 6200 (3,9): 1488 (3,10): 496

trapping sets H_Z (1,3)×124 (2,3)×124 (3,3)×124 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 124 (1,4): 124 (2,3): 124 (2,4): 372 (2,5): 1240 (2,6): 744 (3,3): 124 (3,4): 1116 (3,5): 2852 (3,6): 10168 (3,7): 14632 (3,8): 6200 (3,9): 1488 (3,10): 496

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C62 x C2 = SmallGroup(124, 4) (order 124), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 248; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 6, 33, 47}, b = {0, 32, 90} 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

[[248,20,13]] — two-block group-algebra code over C62 x C2

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| = 248

Parameters are also spelled out in provenance.construction of codes/248-20-13.json.

Parity checks

X-checks 124 (max weight 7) · Z-checks 124 (max weight 7)

H_X (124 checks, sparse supports)

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

H_Z (124 checks, sparse supports)

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

Code ID 248-20-13 · download JSON · raw on GitHub