← back to the board
[[258,32,22]] d ≤
n
258
k
32
d
22
kd²/n
60.031
w
10

Share this result

Distance

d_X 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[26, 48, 55, 60, 64, 77, 100, 101, 109, 113, 138, 151, 155, 165, 177, 183, 190, 191, 192, 236, 248, 255]
d_Z 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[12, 26, 31, 48, 55, 65, 70, 104, 113, 114, 117, 132, 148, 149, 153, 158, 181, 194, 199, 201, 218, 242]
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_129), circulant size 129, a(x) = 1 + x2 + x41 + x73 + x83, b(x) = 1 + x47 + x92 + x112 + x125. Weight-5 supports designed as multiples of a degree-16 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 [22, 22, 22, 22]; 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 (920001-920004) x 1.5M trials/side (gf2_fast), lightest logical 22 each time, nothing lighter; independent BP+OSD corroboration found nothing below 24. Upper bound only. Lit-check 2026-07-14: no published [[258,32,22]] found (closest published GB neighbor [[254,28,14]], arXiv:1904.02703); 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

[[258,32,22]] — designed-divisor weight-10 GB on Z_129

Direction & hypothesis

Target: the weight-9plus × unrestricted cell, which held a single code ([[126,28,8]], kd²/n 14.2). Hypothesis: support-5 GB codes open the cell, but random sampling wastes the weight budget — aligning k with d needs structure.

What was searched

Two tracks. (a) Random support-5 2BGA over 15 nonabelian groups: 85,828 candidates — k and d never aligned; nothing board-advancing survived. (b) The designed-divisor construction that produced this code: pick a divisor g(x) | x^N−1 of target degree (k = 2·deg g guaranteed), then meet-in-the-middle enumerate weight-5 multiples of g. 13,200 such codes screened over cyclic groups N = 63–147 at 1.5k RIS trials. This code: N=129, deg g = 16, a(x) = 1+x²+x⁴¹+x⁷³+x⁸³, b(x) = 1+x⁴⁷+x⁹²+x¹¹²+x¹²⁵.

Evidence trail

Screen values in this family inflate up to ~50% (e.g. 42→34, 38→28, 30→20 by 20k trials), so every survivor was laddered. This code: 8k → 60k → 200k → 1M trials/side all find lightest logical 22, flat from the first rung; claimed d equals the witnessed weight-22 logicals at the 1M rung. Post-hoc adversarial re-check: 4 fresh seeds × 1.5M trials/side, all find 22, nothing lighter. Independent BP+OSD decoder search: nothing below weight 24. Claim: upper bound d ≤ 22, no lighter logical proven absent.

Dead ends

  • Random support-5 (85,828 samples): k·d never beat the designed route — the

    • divisor trick, not sampling volume, was the win.

  • Ten flat-ladder survivors were found; seven remain unpackaged (staged) —

    • confirmation, not generation, is the bottleneck.

  • Nearest literature neighbor checked before claiming: [[254,28,14]]

    • (arXiv:1904.02703); weight-10 supports sit outside the exhaustive W≤8 enumerations. Novelty vs literature: unverified, per board policy.

Tools

Claude Fable 5 multi-agent campaign on the repo's research/ kit; gf2_fast for all deep RIS runs (~9–30s per 1M trials/side at this n); BP+OSD via the decode/ stack for the independent check. ~45 min generation, ~hours of confirmation across parallel agents.

Reproduction

research/kit/group_algebra.py::build_2bga on Z_129 with the supports above (exponents of a(x), b(x)); or see the construction string in codes/258-32-22.json. Code-capacity performance study (BP+OSD, MWPM comparison): research/candidates/campaign-20260714/plots/.

Parity checks

X-checks 129 · Z-checks 129
H_X (129 checks, sparse supports)
[0, 46, 56, 88, 127, 129, 133, 146, 166, 211] [1, 47, 57, 89, 128, 130, 134, 147, 167, 212] [0, 2, 48, 58, 90, 131, 135, 148, 168, 213] [1, 3, 49, 59, 91, 132, 136, 149, 169, 214] [2, 4, 50, 60, 92, 133, 137, 150, 170, 215] [3, 5, 51, 61, 93, 134, 138, 151, 171, 216] [4, 6, 52, 62, 94, 135, 139, 152, 172, 217] [5, 7, 53, 63, 95, 136, 140, 153, 173, 218] [6, 8, 54, 64, 96, 137, 141, 154, 174, 219] [7, 9, 55, 65, 97, 138, 142, 155, 175, 220] [8, 10, 56, 66, 98, 139, 143, 156, 176, 221] [9, 11, 57, 67, 99, 140, 144, 157, 177, 222] [10, 12, 58, 68, 100, 141, 145, 158, 178, 223] [11, 13, 59, 69, 101, 142, 146, 159, 179, 224] [12, 14, 60, 70, 102, 143, 147, 160, 180, 225] [13, 15, 61, 71, 103, 144, 148, 161, 181, 226] [14, 16, 62, 72, 104, 145, 149, 162, 182, 227] [15, 17, 63, 73, 105, 146, 150, 163, 183, 228] [16, 18, 64, 74, 106, 147, 151, 164, 184, 229] [17, 19, 65, 75, 107, 148, 152, 165, 185, 230] [18, 20, 66, 76, 108, 149, 153, 166, 186, 231] [19, 21, 67, 77, 109, 150, 154, 167, 187, 232] [20, 22, 68, 78, 110, 151, 155, 168, 188, 233] [21, 23, 69, 79, 111, 152, 156, 169, 189, 234] [22, 24, 70, 80, 112, 153, 157, 170, 190, 235] [23, 25, 71, 81, 113, 154, 158, 171, 191, 236] [24, 26, 72, 82, 114, 155, 159, 172, 192, 237] [25, 27, 73, 83, 115, 156, 160, 173, 193, 238] [26, 28, 74, 84, 116, 157, 161, 174, 194, 239] [27, 29, 75, 85, 117, 158, 162, 175, 195, 240] [28, 30, 76, 86, 118, 159, 163, 176, 196, 241] [29, 31, 77, 87, 119, 160, 164, 177, 197, 242] [30, 32, 78, 88, 120, 161, 165, 178, 198, 243] [31, 33, 79, 89, 121, 162, 166, 179, 199, 244] [32, 34, 80, 90, 122, 163, 167, 180, 200, 245] [33, 35, 81, 91, 123, 164, 168, 181, 201, 246] [34, 36, 82, 92, 124, 165, 169, 182, 202, 247] [35, 37, 83, 93, 125, 166, 170, 183, 203, 248] [36, 38, 84, 94, 126, 167, 171, 184, 204, 249] [37, 39, 85, 95, 127, 168, 172, 185, 205, 250] [38, 40, 86, 96, 128, 169, 173, 186, 206, 251] [0, 39, 41, 87, 97, 170, 174, 187, 207, 252] [1, 40, 42, 88, 98, 171, 175, 188, 208, 253] [2, 41, 43, 89, 99, 172, 176, 189, 209, 254] [3, 42, 44, 90, 100, 173, 177, 190, 210, 255] [4, 43, 45, 91, 101, 174, 178, 191, 211, 256] [5, 44, 46, 92, 102, 175, 179, 192, 212, 257] [6, 45, 47, 93, 103, 129, 176, 180, 193, 213] [7, 46, 48, 94, 104, 130, 177, 181, 194, 214] [8, 47, 49, 95, 105, 131, 178, 182, 195, 215] [9, 48, 50, 96, 106, 132, 179, 183, 196, 216] [10, 49, 51, 97, 107, 133, 180, 184, 197, 217] [11, 50, 52, 98, 108, 134, 181, 185, 198, 218] [12, 51, 53, 99, 109, 135, 182, 186, 199, 219] [13, 52, 54, 100, 110, 136, 183, 187, 200, 220] [14, 53, 55, 101, 111, 137, 184, 188, 201, 221] [15, 54, 56, 102, 112, 138, 185, 189, 202, 222] [16, 55, 57, 103, 113, 139, 186, 190, 203, 223] [17, 56, 58, 104, 114, 140, 187, 191, 204, 224] [18, 57, 59, 105, 115, 141, 188, 192, 205, 225] [19, 58, 60, 106, 116, 142, 189, 193, 206, 226] [20, 59, 61, 107, 117, 143, 190, 194, 207, 227] [21, 60, 62, 108, 118, 144, 191, 195, 208, 228] [22, 61, 63, 109, 119, 145, 192, 196, 209, 229] [23, 62, 64, 110, 120, 146, 193, 197, 210, 230] [24, 63, 65, 111, 121, 147, 194, 198, 211, 231] [25, 64, 66, 112, 122, 148, 195, 199, 212, 232] [26, 65, 67, 113, 123, 149, 196, 200, 213, 233] [27, 66, 68, 114, 124, 150, 197, 201, 214, 234] [28, 67, 69, 115, 125, 151, 198, 202, 215, 235] [29, 68, 70, 116, 126, 152, 199, 203, 216, 236] [30, 69, 71, 117, 127, 153, 200, 204, 217, 237] [31, 70, 72, 118, 128, 154, 201, 205, 218, 238] [0, 32, 71, 73, 119, 155, 202, 206, 219, 239] [1, 33, 72, 74, 120, 156, 203, 207, 220, 240] [2, 34, 73, 75, 121, 157, 204, 208, 221, 241] [3, 35, 74, 76, 122, 158, 205, 209, 222, 242] [4, 36, 75, 77, 123, 159, 206, 210, 223, 243] [5, 37, 76, 78, 124, 160, 207, 211, 224, 244] [6, 38, 77, 79, 125, 161, 208, 212, 225, 245] [7, 39, 78, 80, 126, 162, 209, 213, 226, 246] [8, 40, 79, 81, 127, 163, 210, 214, 227, 247] [9, 41, 80, 82, 128, 164, 211, 215, 228, 248] [0, 10, 42, 81, 83, 165, 212, 216, 229, 249] [1, 11, 43, 82, 84, 166, 213, 217, 230, 250] [2, 12, 44, 83, 85, 167, 214, 218, 231, 251] [3, 13, 45, 84, 86, 168, 215, 219, 232, 252] [4, 14, 46, 85, 87, 169, 216, 220, 233, 253] [5, 15, 47, 86, 88, 170, 217, 221, 234, 254] [6, 16, 48, 87, 89, 171, 218, 222, 235, 255] [7, 17, 49, 88, 90, 172, 219, 223, 236, 256] [8, 18, 50, 89, 91, 173, 220, 224, 237, 257] [9, 19, 51, 90, 92, 129, 174, 221, 225, 238] [10, 20, 52, 91, 93, 130, 175, 222, 226, 239] [11, 21, 53, 92, 94, 131, 176, 223, 227, 240] [12, 22, 54, 93, 95, 132, 177, 224, 228, 241] [13, 23, 55, 94, 96, 133, 178, 225, 229, 242] [14, 24, 56, 95, 97, 134, 179, 226, 230, 243] [15, 25, 57, 96, 98, 135, 180, 227, 231, 244] [16, 26, 58, 97, 99, 136, 181, 228, 232, 245] [17, 27, 59, 98, 100, 137, 182, 229, 233, 246] [18, 28, 60, 99, 101, 138, 183, 230, 234, 247] [19, 29, 61, 100, 102, 139, 184, 231, 235, 248] [20, 30, 62, 101, 103, 140, 185, 232, 236, 249] [21, 31, 63, 102, 104, 141, 186, 233, 237, 250] [22, 32, 64, 103, 105, 142, 187, 234, 238, 251] [23, 33, 65, 104, 106, 143, 188, 235, 239, 252] [24, 34, 66, 105, 107, 144, 189, 236, 240, 253] [25, 35, 67, 106, 108, 145, 190, 237, 241, 254] [26, 36, 68, 107, 109, 146, 191, 238, 242, 255] [27, 37, 69, 108, 110, 147, 192, 239, 243, 256] [28, 38, 70, 109, 111, 148, 193, 240, 244, 257] [29, 39, 71, 110, 112, 129, 149, 194, 241, 245] [30, 40, 72, 111, 113, 130, 150, 195, 242, 246] [31, 41, 73, 112, 114, 131, 151, 196, 243, 247] [32, 42, 74, 113, 115, 132, 152, 197, 244, 248] [33, 43, 75, 114, 116, 133, 153, 198, 245, 249] [34, 44, 76, 115, 117, 134, 154, 199, 246, 250] [35, 45, 77, 116, 118, 135, 155, 200, 247, 251] [36, 46, 78, 117, 119, 136, 156, 201, 248, 252] [37, 47, 79, 118, 120, 137, 157, 202, 249, 253] [38, 48, 80, 119, 121, 138, 158, 203, 250, 254] [39, 49, 81, 120, 122, 139, 159, 204, 251, 255] [40, 50, 82, 121, 123, 140, 160, 205, 252, 256] [41, 51, 83, 122, 124, 141, 161, 206, 253, 257] [42, 52, 84, 123, 125, 129, 142, 162, 207, 254] [43, 53, 85, 124, 126, 130, 143, 163, 208, 255] [44, 54, 86, 125, 127, 131, 144, 164, 209, 256] [45, 55, 87, 126, 128, 132, 145, 165, 210, 257]
H_Z (129 checks, sparse supports)
[0, 47, 92, 112, 125, 129, 131, 170, 202, 212] [1, 48, 93, 113, 126, 130, 132, 171, 203, 213] [2, 49, 94, 114, 127, 131, 133, 172, 204, 214] [3, 50, 95, 115, 128, 132, 134, 173, 205, 215] [0, 4, 51, 96, 116, 133, 135, 174, 206, 216] [1, 5, 52, 97, 117, 134, 136, 175, 207, 217] [2, 6, 53, 98, 118, 135, 137, 176, 208, 218] [3, 7, 54, 99, 119, 136, 138, 177, 209, 219] [4, 8, 55, 100, 120, 137, 139, 178, 210, 220] [5, 9, 56, 101, 121, 138, 140, 179, 211, 221] [6, 10, 57, 102, 122, 139, 141, 180, 212, 222] [7, 11, 58, 103, 123, 140, 142, 181, 213, 223] [8, 12, 59, 104, 124, 141, 143, 182, 214, 224] [9, 13, 60, 105, 125, 142, 144, 183, 215, 225] [10, 14, 61, 106, 126, 143, 145, 184, 216, 226] [11, 15, 62, 107, 127, 144, 146, 185, 217, 227] [12, 16, 63, 108, 128, 145, 147, 186, 218, 228] [0, 13, 17, 64, 109, 146, 148, 187, 219, 229] [1, 14, 18, 65, 110, 147, 149, 188, 220, 230] [2, 15, 19, 66, 111, 148, 150, 189, 221, 231] [3, 16, 20, 67, 112, 149, 151, 190, 222, 232] [4, 17, 21, 68, 113, 150, 152, 191, 223, 233] [5, 18, 22, 69, 114, 151, 153, 192, 224, 234] [6, 19, 23, 70, 115, 152, 154, 193, 225, 235] [7, 20, 24, 71, 116, 153, 155, 194, 226, 236] [8, 21, 25, 72, 117, 154, 156, 195, 227, 237] [9, 22, 26, 73, 118, 155, 157, 196, 228, 238] [10, 23, 27, 74, 119, 156, 158, 197, 229, 239] [11, 24, 28, 75, 120, 157, 159, 198, 230, 240] [12, 25, 29, 76, 121, 158, 160, 199, 231, 241] [13, 26, 30, 77, 122, 159, 161, 200, 232, 242] [14, 27, 31, 78, 123, 160, 162, 201, 233, 243] [15, 28, 32, 79, 124, 161, 163, 202, 234, 244] [16, 29, 33, 80, 125, 162, 164, 203, 235, 245] [17, 30, 34, 81, 126, 163, 165, 204, 236, 246] [18, 31, 35, 82, 127, 164, 166, 205, 237, 247] [19, 32, 36, 83, 128, 165, 167, 206, 238, 248] [0, 20, 33, 37, 84, 166, 168, 207, 239, 249] [1, 21, 34, 38, 85, 167, 169, 208, 240, 250] [2, 22, 35, 39, 86, 168, 170, 209, 241, 251] [3, 23, 36, 40, 87, 169, 171, 210, 242, 252] [4, 24, 37, 41, 88, 170, 172, 211, 243, 253] [5, 25, 38, 42, 89, 171, 173, 212, 244, 254] [6, 26, 39, 43, 90, 172, 174, 213, 245, 255] [7, 27, 40, 44, 91, 173, 175, 214, 246, 256] [8, 28, 41, 45, 92, 174, 176, 215, 247, 257] [9, 29, 42, 46, 93, 129, 175, 177, 216, 248] [10, 30, 43, 47, 94, 130, 176, 178, 217, 249] [11, 31, 44, 48, 95, 131, 177, 179, 218, 250] [12, 32, 45, 49, 96, 132, 178, 180, 219, 251] [13, 33, 46, 50, 97, 133, 179, 181, 220, 252] [14, 34, 47, 51, 98, 134, 180, 182, 221, 253] [15, 35, 48, 52, 99, 135, 181, 183, 222, 254] [16, 36, 49, 53, 100, 136, 182, 184, 223, 255] [17, 37, 50, 54, 101, 137, 183, 185, 224, 256] [18, 38, 51, 55, 102, 138, 184, 186, 225, 257] [19, 39, 52, 56, 103, 129, 139, 185, 187, 226] [20, 40, 53, 57, 104, 130, 140, 186, 188, 227] [21, 41, 54, 58, 105, 131, 141, 187, 189, 228] [22, 42, 55, 59, 106, 132, 142, 188, 190, 229] [23, 43, 56, 60, 107, 133, 143, 189, 191, 230] [24, 44, 57, 61, 108, 134, 144, 190, 192, 231] [25, 45, 58, 62, 109, 135, 145, 191, 193, 232] [26, 46, 59, 63, 110, 136, 146, 192, 194, 233] [27, 47, 60, 64, 111, 137, 147, 193, 195, 234] [28, 48, 61, 65, 112, 138, 148, 194, 196, 235] [29, 49, 62, 66, 113, 139, 149, 195, 197, 236] [30, 50, 63, 67, 114, 140, 150, 196, 198, 237] [31, 51, 64, 68, 115, 141, 151, 197, 199, 238] [32, 52, 65, 69, 116, 142, 152, 198, 200, 239] [33, 53, 66, 70, 117, 143, 153, 199, 201, 240] [34, 54, 67, 71, 118, 144, 154, 200, 202, 241] [35, 55, 68, 72, 119, 145, 155, 201, 203, 242] [36, 56, 69, 73, 120, 146, 156, 202, 204, 243] [37, 57, 70, 74, 121, 147, 157, 203, 205, 244] [38, 58, 71, 75, 122, 148, 158, 204, 206, 245] [39, 59, 72, 76, 123, 149, 159, 205, 207, 246] [40, 60, 73, 77, 124, 150, 160, 206, 208, 247] [41, 61, 74, 78, 125, 151, 161, 207, 209, 248] [42, 62, 75, 79, 126, 152, 162, 208, 210, 249] [43, 63, 76, 80, 127, 153, 163, 209, 211, 250] [44, 64, 77, 81, 128, 154, 164, 210, 212, 251] [0, 45, 65, 78, 82, 155, 165, 211, 213, 252] [1, 46, 66, 79, 83, 156, 166, 212, 214, 253] [2, 47, 67, 80, 84, 157, 167, 213, 215, 254] [3, 48, 68, 81, 85, 158, 168, 214, 216, 255] [4, 49, 69, 82, 86, 159, 169, 215, 217, 256] [5, 50, 70, 83, 87, 160, 170, 216, 218, 257] [6, 51, 71, 84, 88, 129, 161, 171, 217, 219] [7, 52, 72, 85, 89, 130, 162, 172, 218, 220] [8, 53, 73, 86, 90, 131, 163, 173, 219, 221] [9, 54, 74, 87, 91, 132, 164, 174, 220, 222] [10, 55, 75, 88, 92, 133, 165, 175, 221, 223] [11, 56, 76, 89, 93, 134, 166, 176, 222, 224] [12, 57, 77, 90, 94, 135, 167, 177, 223, 225] [13, 58, 78, 91, 95, 136, 168, 178, 224, 226] [14, 59, 79, 92, 96, 137, 169, 179, 225, 227] [15, 60, 80, 93, 97, 138, 170, 180, 226, 228] [16, 61, 81, 94, 98, 139, 171, 181, 227, 229] [17, 62, 82, 95, 99, 140, 172, 182, 228, 230] [18, 63, 83, 96, 100, 141, 173, 183, 229, 231] [19, 64, 84, 97, 101, 142, 174, 184, 230, 232] [20, 65, 85, 98, 102, 143, 175, 185, 231, 233] [21, 66, 86, 99, 103, 144, 176, 186, 232, 234] [22, 67, 87, 100, 104, 145, 177, 187, 233, 235] [23, 68, 88, 101, 105, 146, 178, 188, 234, 236] [24, 69, 89, 102, 106, 147, 179, 189, 235, 237] [25, 70, 90, 103, 107, 148, 180, 190, 236, 238] [26, 71, 91, 104, 108, 149, 181, 191, 237, 239] [27, 72, 92, 105, 109, 150, 182, 192, 238, 240] [28, 73, 93, 106, 110, 151, 183, 193, 239, 241] [29, 74, 94, 107, 111, 152, 184, 194, 240, 242] [30, 75, 95, 108, 112, 153, 185, 195, 241, 243] [31, 76, 96, 109, 113, 154, 186, 196, 242, 244] [32, 77, 97, 110, 114, 155, 187, 197, 243, 245] [33, 78, 98, 111, 115, 156, 188, 198, 244, 246] [34, 79, 99, 112, 116, 157, 189, 199, 245, 247] [35, 80, 100, 113, 117, 158, 190, 200, 246, 248] [36, 81, 101, 114, 118, 159, 191, 201, 247, 249] [37, 82, 102, 115, 119, 160, 192, 202, 248, 250] [38, 83, 103, 116, 120, 161, 193, 203, 249, 251] [39, 84, 104, 117, 121, 162, 194, 204, 250, 252] [40, 85, 105, 118, 122, 163, 195, 205, 251, 253] [41, 86, 106, 119, 123, 164, 196, 206, 252, 254] [42, 87, 107, 120, 124, 165, 197, 207, 253, 255] [43, 88, 108, 121, 125, 166, 198, 208, 254, 256] [44, 89, 109, 122, 126, 167, 199, 209, 255, 257] [45, 90, 110, 123, 127, 129, 168, 200, 210, 256] [46, 91, 111, 124, 128, 130, 169, 201, 211, 257]
raw submission JSON