← back to the board
[[254,44,21]] d ≤
n
254
k
44
d
21
kd²/n
76.394
w
12

Share this result

Distance

d_X 21 · witness weight 21 (claimed upper_bound)
witness operator (support, 21 qubits)
[9, 12, 13, 32, 41, 54, 58, 69, 86, 98, 103, 107, 122, 129, 176, 220, 224, 231, 233, 242, 252]
d_Z 21 · witness weight 21 (claimed upper_bound)
witness operator (support, 21 qubits)
[26, 73, 77, 87, 96, 98, 105, 109, 133, 144, 148, 161, 170, 189, 190, 193, 207, 222, 226, 231, 243]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Generalized bicycle (2BGA on Z_127), circulant size 127, a(x) = 1 + x10 + x12 + x25 + x88 + x95, b(x) = 1 + x + x29 + x37 + x96 + x119. Weight-6 supports designed as multiples of a degree-22 divisor g of x127 - 1 (g = (x-1) times three degree-7 irreducibles; k = 2 deg g = 44). Check weight 12 (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-6 GB campaign 20260714 (divisor design over Z_127). Screen 250 RIS trials across all 816 degree-22 divisors of x^127 - 1; refine top candidates at 8000; ladder [60000, 1000000] -> d [21, 21]; claimed d = lightest logical witnessed (gf2_fast RIS). Independent adversarial re-check: 4 fresh seeds (940001-940004) x 1.5M trials (gf2_fast), lightest logical 21 each time, nothing lighter. Independent BP+OSD syndrome-decoder cross-check (60000 trials) corroborated d=21, nothing lighter. Upper bound only. Lit-check 2026-07-14: no published [[254,44,21]] found; closest published n=254 GB neighbor [[254,28,14]] (arXiv:1904.02703); weight-12 supports are outside the published 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

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

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