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[[210,60,8]] d ≤
n
210
k
60
d
8
kd²/n
18.286
w
25
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · w_X = 25, w_Z = 25 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[49, 51, 61, 63, 113, 115, 125, 127]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[49, 51, 61, 63, 113, 115, 125, 127]
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 5–25 (mean 9.857) · H_Z 5–25 (mean 9.857)
qubit degrees H_X 2–24 (mean 3.943) · H_Z 2–24 (mean 3.943)
trapping sets H_X (1,2)×24 (2,2)×36 (3,2)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 24 (1,3): 72 (1,4): 81 (1,5): 24 (1,8): 6 (1,24): 3 (2,2): 36 (2,3): 288 (2,4): 840 (2,5): 1320 (2,6): 1044 (2,7): 96 (2,8): 60 (2,9): 72 (2,10): 72 (2,11): 24 (2,24): 93 (2,25): 72 (2,26): 144 (3,2): 36 (3,3): 216 (3,4): 4632 (3,5): 12096 (3,6): 18096 (3,7): 23088 (3,8): 18816 (3,9): 8840 (3,10): 2664 (3,11): 3024 (3,12): 2196 (3,13): 336 (3,14): 108 (3,16): 36 (3,23): 216 (3,24): 1900 (3,25): 3456 (3,26): 5904 (3,27): 4608 (3,28): 4212 (3,30): 216
trapping sets H_Z (1,2)×24 (2,2)×36 (3,2)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 24 (1,3): 72 (1,4): 81 (1,5): 24 (1,8): 6 (1,24): 3 (2,2): 36 (2,3): 288 (2,4): 840 (2,5): 1320 (2,6): 1044 (2,7): 96 (2,8): 60 (2,9): 72 (2,10): 72 (2,11): 24 (2,24): 93 (2,25): 72 (2,26): 144 (3,2): 36 (3,3): 216 (3,4): 4632 (3,5): 12096 (3,6): 18096 (3,7): 23088 (3,8): 18816 (3,9): 8840 (3,10): 2664 (3,11): 3024 (3,12): 2196 (3,13): 336 (3,14): 108 (3,16): 36 (3,23): 216 (3,24): 1900 (3,25): 3456 (3,26): 5904 (3,27): 4608 (3,28): 4212 (3,30): 216

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction One amplification of [[48,30,4]] base by flagged [[4,2,2]] amplifier (arXiv:2609.37231).
model Beam 1.0 (claimed, not verified)
date 2026-10-18
family other (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

[[210,60,8]] — amplification of [[48,30,4]] by the flagged [[4,2,2]] amplifier

Direction & hypothesis

Target: unrestricted / any-weight cell. One amplification step of the [[48,30,4]] base by the flagged [[4,2,2]] amplifier (arXiv:2609.37231).

What was searched

Single construction; board domination check found no dominator.

Evidence trail

d=8 upper bound (4*2). CLI RIS search passed.

Tools

Model: Beam 1.0. Base code: codes/48-30-4.json.

Reproduction

Base: codes/48-30-4.json. Amplifier: flagged [[4,2,2]]. Central three-term tensor product. Result n=210, k=60, d=8, w=25.

Parity checks

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