[[220,52,5]] · QEC Challenge

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[[220,52,5]] d =

220

kd²/n

5.909

X/Z

0.0058

5.6569

layers

swaps

977

Share this result

Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 5 · witness weight 5 (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, 5 qubits)

[30, 41, 52, 63, 74]

d_Z 5 · witness weight 5 (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, 5 qubits)

[2, 7, 14, 23, 34]

certificate exact, d = 5 · CryptoMiniSat 5.14.7 SAT

X: no logical < 5 exists; Z: no logical < 5 exists

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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×220 (2,4)×440 (3,4)×1320 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 220 (2,4): 440 (2,6): 2200 (3,4): 1320 (3,6): 8800 (3,8): 33000 (3,10): 1760

trapping sets H_Z (1,4)×220 (2,4)×440 (3,4)×1320 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 220 (2,4): 440 (2,6): 2200 (3,4): 1320 (3,6): 8800 (3,8): 33000 (3,10): 1760

witness diameter X 3.6056 · Z 3.0 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

layout contributed by @MathysRennela · Simulated annealing over integer-grid site assignments with the repo's own research/local2d/fold_layout.search_layout, single layer at radius 4 then two layers at 7, plus a divisor scan of the two-block index convention Z_l x Z_m which gives the tighter embedding here; radius and per-site occupancy re-measured with the verifier's own rule. · 2026-10-13

X checkZ checkqubit site (112)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

routing cost 977 nearest-neighbor SWAPs per round in total, at most 10 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors Lee, Gideon

provenance literature baseline

construction Two-block group-algebra code C(a,b) over C11 x D10 = SmallGroup(110, 3) (order 110), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 220; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 8, 30, 52}, b = {0, 3, 9, 17} as element indices. Check weight 8. 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 2D-local bilayer (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

[[220,52,5]] — two-block group-algebra code over C11 x D10

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| = 220

Parameters are also spelled out in provenance.construction of codes/220-52-5.json.

Parity checks

X-checks 110 (max weight 8) · Z-checks 110 (max weight 8)

H_X (110 checks, sparse supports)

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

H_Z (110 checks, sparse supports)

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

Code ID 220-52-5 · download JSON · raw on GitHub