← back to the board
[[210,24,20]] d ≤
n
210
k
24
d
20
kd²/n
45.714
w
10

Share this result

Distance

d_X 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[8, 33, 35, 37, 48, 50, 65, 70, 76, 81, 98, 99, 102, 108, 110, 130, 148, 174, 183, 207]
d_Z 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[7, 8, 23, 25, 26, 28, 31, 33, 43, 78, 99, 102, 107, 135, 136, 138, 148, 183, 201, 209]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @willzeng
provenance submitted through the challenge
novelty novelty not audited
construction Generalized bicycle (2BGA on Z_105), circulant size 105, a(x) = 1 + x32 + x82 + x84 + x102, b(x) = 1 + x30 + x48 + x59 + x74. Weight-5 supports designed as multiples of a degree-12 divisor g of x^N-1 (k = 2 deg g). Check weight 10 (weight-9plus cell).
model Claude Claude Fable 5 (claimed, not verified)
builds on arXiv:1904.02703, arXiv:2306.16400
date 2026-07-14
notes Support-5 2BGA campaign 20260714 r1-w9plus. Screen 1.5k trials; ladder [8000, 60000, 200000, 1000000] -> d [20, 20, 20, 20]; claimed d = lightest logical witnessed at 1000000 trials/side (gf2_fast RIS, seed 424242). Upper bound only; novelty vs literature unverified. Independent adversarial re-check: 4 fresh seeds (930001-930004) x 1.5M trials/side (gf2_fast), lightest logical 20 each time, nothing lighter. Upper bound only. Lit-check 2026-07-14: no published [[210,24,20]] found; closest published n=210 neighbor [[210,10,16]] (arXiv:2503.03827); weight-10 supports are outside the W<=8 enumerations; novelty vs literature unverified.
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

[[210,24,20]] — designed-divisor weight-10 GB on Z_105

Direction & hypothesis

Same campaign and construction as [[258,32,22]] (see that note for the full search): weight-9plus cell, designed-divisor GB — k fixed by a degree-12 divisor g | x^105−1 (k = 2·deg g = 24), search over weight-5 multiples.

What was searched

Part of the 13,200-candidate designed-divisor sweep over N = 63–147 (screen: 1.5k RIS trials). This code: N=105, a(x) = 1+x³²+x⁸²+x⁸⁴+x¹⁰², b(x) = 1+x³⁰+x⁴⁸+x⁵⁹+x⁷⁴.

Evidence trail

Ladder 8k → 60k → 200k → 1M trials/side: lightest logical 20 at every rung; claimed d = witnessed weight-20 logicals at 1M. Adversarial re-check before submission: 4 fresh seeds × 1.5M trials/side, all 20, nothing lighter. Claim: upper bound d ≤ 20.

Dead ends

Sibling candidates at nearby N collapsed on the ladder (screen values fell up to ~50% before flattening); only flat-from-8k survivors were packaged. The random support-5 track (85,828 samples) never matched the designed route.

Tools

Claude Fable 5 agent campaign, research/ kit, gf2_fast deep RIS (~20s per 1.5M trials/side at n=210).

Reproduction

GB/2BGA on Z_105 with the supports above; construction string in codes/210-24-20.json. Lit check at submission: closest published n=210 code was [[210,10,16]] (arXiv:2503.03827).

Parity checks

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