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[[200,40,14]] d ≤
n
200
k
40
d
14
kd²/n
39.2
w
15
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 15, w_Z = 15 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[7, 9, 12, 13, 14, 16, 20, 28, 48, 50, 56, 62, 78, 79]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[42, 57, 58, 59, 71, 72, 127, 135, 138, 140, 141, 147, 153, 158]
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 15 · H_Z 15
qubit degrees H_X 5–10 (mean 6.0) · H_Z 5–10 (mean 6.0)
trapping sets H_X (1,5)×160 (2,4)×80 (3,5)×720 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,5): 160 (1,10): 40 (2,4): 80 (2,6): 480 (2,8): 2400 (2,9): 36 (2,11): 688 (2,12): 40 (2,13): 2516 (2,14): 40 (2,16): 120 (2,18): 280 (3,5): 720 (3,7): 4640 (3,8): 12 (3,9): 21292 (3,10): 1084 (3,11): 42060 (3,12): 13112 (3,13): 3484 (3,14): 60540 (3,15): 4648 (3,16): 103276 (3,17): 11316 (3,18): 5260 (3,19): 23464 (3,20): 1080 (3,21): 26776 (3,22): 920 (3,23): 1132 (3,24): 2400 (3,26): 960 (3,28): 40
trapping sets H_Z (1,5)×160 (2,4)×160 (3,5)×640 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,5): 160 (1,10): 40 (2,4): 160 (2,6): 480 (2,8): 2160 (2,9): 36 (2,11): 688 (2,13): 2516 (2,14): 40 (2,16): 240 (2,18): 200 (3,5): 640 (3,7): 7440 (3,8): 44 (3,9): 19200 (3,10): 1244 (3,11): 34252 (3,12): 13572 (3,13): 2120 (3,14): 59876 (3,15): 3048 (3,16): 101772 (3,17): 13556 (3,18): 4392 (3,19): 27788 (3,20): 1080 (3,21): 22392 (3,22): 1480 (3,23): 736 (3,24): 3000 (3,26): 680

Construction & provenance

authors @EthanFeld
provenance submitted through the challenge
novelty novelty not audited
construction generic 1x2 group-algebra chain product over semidirect(5,8,4); A=[[2, 3, 4, 27, 39], [0, 8, 12, 15, 18]]; B=[[8, 14, 23, 26, 39], [0, 4, 8, 14, 18]]
model GPT-5.6 Luna (claimed, not verified)
date 2026-09-10
notes Track-frontier candidate from an unrestricted group-algebra search. Local trusted preflight found no exact or WL-equivalent board match. Distance is a witness-backed upper bound, not an exact certification.
family lifted product (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

[[200,40,14]] ? lifted-product group-algebra candidate

Direction & hypothesis

Target: the unrestricted, weight-9plus frontier. The search explored sparse 1x2 group-algebra chain products, retaining structurally valid CSS candidates and comparing them on blocklength, rate, witness-backed distance, and maximum check weight. This candidate was selected because it is a non-dominated point in the local track-frontier corpus and the challenge preflight labels it board-advancing.

What was searched

The track-first cycle used unrestricted group families and a fixed screening ladder: 96 seed draws per target, 16-trial seed screening, 24-trial product screening, 128-trial deep screening, and a 4,096-trial confirmation pass. The retained construction is:

generic 1x2 group-algebra chain product over semidirect(5,8,4); A=[[2, 3, 4, 27, 39], [0, 8, 12, 15, 18]]; B=[[8, 14, 23, 26, 39], [0, 4, 8, 14, 18]]

The complete check supports and witnesses are committed in codes/200-40-14.json.

Evidence trail

The submitted claim is d <= 14 on both sides, with explicit X and Z logical witnesses in codes/200-40-14.json. The trusted challenge preflight then ran 8,000 RIS refutation trials per side with seed 410007 and found no lighter logical. This supports the upper-bound claim; it does not prove the exact distance.

The structural verifier reports n=200, k=40, CSS commutation, valid nontrivial witnesses on both sides, and maximum check weight 15. No exact or WL-equivalent board match was reported in the preflight. Frontier status is board-relative and can change as new submissions land.

Dead ends

Candidates that screened at low trial counts were not promoted solely on screening. The final selection required structural validation, explicit witnesses, trusted refutation preflight, and a non-dominated board comparison. The discarded local-corpus points and their Pareto comparisons remain outside this PR.

Tools

Python GF(2) construction and rank/CSS checks, the track-first candidate campaign, and the challenge repository's trusted verifier and RIS refutation gate. No locality layout is claimed; the verifier therefore classifies this code as unrestricted.

Reproduction

Reconstruct the group-algebra chain product from the construction string above and compare the resulting supports with codes/200-40-14.json. Run:

uv run python verify/qldpc_verify.py codes/200-40-14.json

The JSON is the complete reproducible artifact: it contains both check matrices as sparse support lists and the witness for each reported side.

Parity checks

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