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[[288,50,8]] d ≤
n
288
k
50
d
8
kd²/n
11.111
w
8

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Distance

d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[72, 79, 80, 87, 220, 223, 260, 263]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[9, 11, 14, 49, 51, 54, 188, 196]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors Cruz-Benito, Juan and Cross, Andrew W. and Kremer, David and Faro, Ismael
provenance literature baseline
construction Bivariate bicycle on Z_18 x Z_8: A(x,y) = (1+x)(1+y5) = 1 + x + y5 + x y5, B(x,y) = (1+y)(1+x5) = 1 + y + x5 + x5 y (weight-8 mixed-monomial, cross-factored). From arXiv:2606.02418 Table III; distance d=8 proven exact there by MILP.
model classical construction (no AI model)
reference arXiv:2606.02418
date 2026-06-01
notes High-rate (k=50) weight-8 BB code seeded from the LLM-guided evolutionary search of Cruz-Benito et al.; kd^2/n = 11.1. Distance witnessed here as d<=8 upper bound; source reports d=8 exact by MILP.
family bivariate 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 144 · Z-checks 144
H_X (144 checks, sparse supports)
[0, 5, 8, 13, 144, 145, 184, 185] [1, 6, 9, 14, 145, 146, 185, 186] [2, 7, 10, 15, 146, 147, 186, 187] [0, 3, 8, 11, 147, 148, 187, 188] [1, 4, 9, 12, 148, 149, 188, 189] [2, 5, 10, 13, 149, 150, 189, 190] [3, 6, 11, 14, 150, 151, 190, 191] [4, 7, 12, 15, 144, 151, 184, 191] [8, 13, 16, 21, 152, 153, 192, 193] [9, 14, 17, 22, 153, 154, 193, 194] [10, 15, 18, 23, 154, 155, 194, 195] [8, 11, 16, 19, 155, 156, 195, 196] [9, 12, 17, 20, 156, 157, 196, 197] [10, 13, 18, 21, 157, 158, 197, 198] [11, 14, 19, 22, 158, 159, 198, 199] [12, 15, 20, 23, 152, 159, 192, 199] [16, 21, 24, 29, 160, 161, 200, 201] [17, 22, 25, 30, 161, 162, 201, 202] [18, 23, 26, 31, 162, 163, 202, 203] [16, 19, 24, 27, 163, 164, 203, 204] [17, 20, 25, 28, 164, 165, 204, 205] [18, 21, 26, 29, 165, 166, 205, 206] [19, 22, 27, 30, 166, 167, 206, 207] [20, 23, 28, 31, 160, 167, 200, 207] [24, 29, 32, 37, 168, 169, 208, 209] [25, 30, 33, 38, 169, 170, 209, 210] [26, 31, 34, 39, 170, 171, 210, 211] [24, 27, 32, 35, 171, 172, 211, 212] [25, 28, 33, 36, 172, 173, 212, 213] [26, 29, 34, 37, 173, 174, 213, 214] [27, 30, 35, 38, 174, 175, 214, 215] [28, 31, 36, 39, 168, 175, 208, 215] [32, 37, 40, 45, 176, 177, 216, 217] [33, 38, 41, 46, 177, 178, 217, 218] [34, 39, 42, 47, 178, 179, 218, 219] [32, 35, 40, 43, 179, 180, 219, 220] [33, 36, 41, 44, 180, 181, 220, 221] [34, 37, 42, 45, 181, 182, 221, 222] [35, 38, 43, 46, 182, 183, 222, 223] [36, 39, 44, 47, 176, 183, 216, 223] [40, 45, 48, 53, 184, 185, 224, 225] [41, 46, 49, 54, 185, 186, 225, 226] [42, 47, 50, 55, 186, 187, 226, 227] [40, 43, 48, 51, 187, 188, 227, 228] [41, 44, 49, 52, 188, 189, 228, 229] [42, 45, 50, 53, 189, 190, 229, 230] [43, 46, 51, 54, 190, 191, 230, 231] [44, 47, 52, 55, 184, 191, 224, 231] [48, 53, 56, 61, 192, 193, 232, 233] [49, 54, 57, 62, 193, 194, 233, 234] [50, 55, 58, 63, 194, 195, 234, 235] [48, 51, 56, 59, 195, 196, 235, 236] [49, 52, 57, 60, 196, 197, 236, 237] [50, 53, 58, 61, 197, 198, 237, 238] [51, 54, 59, 62, 198, 199, 238, 239] [52, 55, 60, 63, 192, 199, 232, 239] [56, 61, 64, 69, 200, 201, 240, 241] [57, 62, 65, 70, 201, 202, 241, 242] [58, 63, 66, 71, 202, 203, 242, 243] [56, 59, 64, 67, 203, 204, 243, 244] [57, 60, 65, 68, 204, 205, 244, 245] [58, 61, 66, 69, 205, 206, 245, 246] [59, 62, 67, 70, 206, 207, 246, 247] [60, 63, 68, 71, 200, 207, 240, 247] [64, 69, 72, 77, 208, 209, 248, 249] [65, 70, 73, 78, 209, 210, 249, 250] [66, 71, 74, 79, 210, 211, 250, 251] [64, 67, 72, 75, 211, 212, 251, 252] [65, 68, 73, 76, 212, 213, 252, 253] [66, 69, 74, 77, 213, 214, 253, 254] [67, 70, 75, 78, 214, 215, 254, 255] [68, 71, 76, 79, 208, 215, 248, 255] [72, 77, 80, 85, 216, 217, 256, 257] [73, 78, 81, 86, 217, 218, 257, 258] [74, 79, 82, 87, 218, 219, 258, 259] [72, 75, 80, 83, 219, 220, 259, 260] [73, 76, 81, 84, 220, 221, 260, 261] [74, 77, 82, 85, 221, 222, 261, 262] [75, 78, 83, 86, 222, 223, 262, 263] [76, 79, 84, 87, 216, 223, 256, 263] [80, 85, 88, 93, 224, 225, 264, 265] [81, 86, 89, 94, 225, 226, 265, 266] [82, 87, 90, 95, 226, 227, 266, 267] [80, 83, 88, 91, 227, 228, 267, 268] [81, 84, 89, 92, 228, 229, 268, 269] [82, 85, 90, 93, 229, 230, 269, 270] [83, 86, 91, 94, 230, 231, 270, 271] [84, 87, 92, 95, 224, 231, 264, 271] [88, 93, 96, 101, 232, 233, 272, 273] [89, 94, 97, 102, 233, 234, 273, 274] [90, 95, 98, 103, 234, 235, 274, 275] [88, 91, 96, 99, 235, 236, 275, 276] [89, 92, 97, 100, 236, 237, 276, 277] [90, 93, 98, 101, 237, 238, 277, 278] [91, 94, 99, 102, 238, 239, 278, 279] [92, 95, 100, 103, 232, 239, 272, 279] [96, 101, 104, 109, 240, 241, 280, 281] [97, 102, 105, 110, 241, 242, 281, 282] [98, 103, 106, 111, 242, 243, 282, 283] [96, 99, 104, 107, 243, 244, 283, 284] [97, 100, 105, 108, 244, 245, 284, 285] [98, 101, 106, 109, 245, 246, 285, 286] [99, 102, 107, 110, 246, 247, 286, 287] [100, 103, 108, 111, 240, 247, 280, 287] [104, 109, 112, 117, 144, 145, 248, 249] [105, 110, 113, 118, 145, 146, 249, 250] [106, 111, 114, 119, 146, 147, 250, 251] [104, 107, 112, 115, 147, 148, 251, 252] [105, 108, 113, 116, 148, 149, 252, 253] [106, 109, 114, 117, 149, 150, 253, 254] [107, 110, 115, 118, 150, 151, 254, 255] [108, 111, 116, 119, 144, 151, 248, 255] [112, 117, 120, 125, 152, 153, 256, 257] [113, 118, 121, 126, 153, 154, 257, 258] [114, 119, 122, 127, 154, 155, 258, 259] [112, 115, 120, 123, 155, 156, 259, 260] [113, 116, 121, 124, 156, 157, 260, 261] [114, 117, 122, 125, 157, 158, 261, 262] [115, 118, 123, 126, 158, 159, 262, 263] [116, 119, 124, 127, 152, 159, 256, 263] [120, 125, 128, 133, 160, 161, 264, 265] [121, 126, 129, 134, 161, 162, 265, 266] [122, 127, 130, 135, 162, 163, 266, 267] [120, 123, 128, 131, 163, 164, 267, 268] [121, 124, 129, 132, 164, 165, 268, 269] [122, 125, 130, 133, 165, 166, 269, 270] [123, 126, 131, 134, 166, 167, 270, 271] [124, 127, 132, 135, 160, 167, 264, 271] [128, 133, 136, 141, 168, 169, 272, 273] [129, 134, 137, 142, 169, 170, 273, 274] [130, 135, 138, 143, 170, 171, 274, 275] [128, 131, 136, 139, 171, 172, 275, 276] [129, 132, 137, 140, 172, 173, 276, 277] [130, 133, 138, 141, 173, 174, 277, 278] [131, 134, 139, 142, 174, 175, 278, 279] [132, 135, 140, 143, 168, 175, 272, 279] [0, 5, 136, 141, 176, 177, 280, 281] [1, 6, 137, 142, 177, 178, 281, 282] [2, 7, 138, 143, 178, 179, 282, 283] [0, 3, 136, 139, 179, 180, 283, 284] [1, 4, 137, 140, 180, 181, 284, 285] [2, 5, 138, 141, 181, 182, 285, 286] [3, 6, 139, 142, 182, 183, 286, 287] [4, 7, 140, 143, 176, 183, 280, 287]
H_Z (144 checks, sparse supports)
[0, 7, 104, 111, 144, 147, 280, 283] [0, 1, 104, 105, 145, 148, 281, 284] [1, 2, 105, 106, 146, 149, 282, 285] [2, 3, 106, 107, 147, 150, 283, 286] [3, 4, 107, 108, 148, 151, 284, 287] [4, 5, 108, 109, 144, 149, 280, 285] [5, 6, 109, 110, 145, 150, 281, 286] [6, 7, 110, 111, 146, 151, 282, 287] [8, 15, 112, 119, 144, 147, 152, 155] [8, 9, 112, 113, 145, 148, 153, 156] [9, 10, 113, 114, 146, 149, 154, 157] [10, 11, 114, 115, 147, 150, 155, 158] [11, 12, 115, 116, 148, 151, 156, 159] [12, 13, 116, 117, 144, 149, 152, 157] [13, 14, 117, 118, 145, 150, 153, 158] [14, 15, 118, 119, 146, 151, 154, 159] [16, 23, 120, 127, 152, 155, 160, 163] [16, 17, 120, 121, 153, 156, 161, 164] [17, 18, 121, 122, 154, 157, 162, 165] [18, 19, 122, 123, 155, 158, 163, 166] [19, 20, 123, 124, 156, 159, 164, 167] [20, 21, 124, 125, 152, 157, 160, 165] [21, 22, 125, 126, 153, 158, 161, 166] [22, 23, 126, 127, 154, 159, 162, 167] [24, 31, 128, 135, 160, 163, 168, 171] [24, 25, 128, 129, 161, 164, 169, 172] [25, 26, 129, 130, 162, 165, 170, 173] [26, 27, 130, 131, 163, 166, 171, 174] [27, 28, 131, 132, 164, 167, 172, 175] [28, 29, 132, 133, 160, 165, 168, 173] [29, 30, 133, 134, 161, 166, 169, 174] [30, 31, 134, 135, 162, 167, 170, 175] [32, 39, 136, 143, 168, 171, 176, 179] [32, 33, 136, 137, 169, 172, 177, 180] [33, 34, 137, 138, 170, 173, 178, 181] [34, 35, 138, 139, 171, 174, 179, 182] [35, 36, 139, 140, 172, 175, 180, 183] [36, 37, 140, 141, 168, 173, 176, 181] [37, 38, 141, 142, 169, 174, 177, 182] [38, 39, 142, 143, 170, 175, 178, 183] [0, 7, 40, 47, 176, 179, 184, 187] [0, 1, 40, 41, 177, 180, 185, 188] [1, 2, 41, 42, 178, 181, 186, 189] [2, 3, 42, 43, 179, 182, 187, 190] [3, 4, 43, 44, 180, 183, 188, 191] [4, 5, 44, 45, 176, 181, 184, 189] [5, 6, 45, 46, 177, 182, 185, 190] [6, 7, 46, 47, 178, 183, 186, 191] [8, 15, 48, 55, 184, 187, 192, 195] [8, 9, 48, 49, 185, 188, 193, 196] [9, 10, 49, 50, 186, 189, 194, 197] [10, 11, 50, 51, 187, 190, 195, 198] [11, 12, 51, 52, 188, 191, 196, 199] [12, 13, 52, 53, 184, 189, 192, 197] [13, 14, 53, 54, 185, 190, 193, 198] [14, 15, 54, 55, 186, 191, 194, 199] [16, 23, 56, 63, 192, 195, 200, 203] [16, 17, 56, 57, 193, 196, 201, 204] [17, 18, 57, 58, 194, 197, 202, 205] [18, 19, 58, 59, 195, 198, 203, 206] [19, 20, 59, 60, 196, 199, 204, 207] [20, 21, 60, 61, 192, 197, 200, 205] [21, 22, 61, 62, 193, 198, 201, 206] [22, 23, 62, 63, 194, 199, 202, 207] [24, 31, 64, 71, 200, 203, 208, 211] [24, 25, 64, 65, 201, 204, 209, 212] [25, 26, 65, 66, 202, 205, 210, 213] [26, 27, 66, 67, 203, 206, 211, 214] [27, 28, 67, 68, 204, 207, 212, 215] [28, 29, 68, 69, 200, 205, 208, 213] [29, 30, 69, 70, 201, 206, 209, 214] [30, 31, 70, 71, 202, 207, 210, 215] [32, 39, 72, 79, 208, 211, 216, 219] [32, 33, 72, 73, 209, 212, 217, 220] [33, 34, 73, 74, 210, 213, 218, 221] [34, 35, 74, 75, 211, 214, 219, 222] [35, 36, 75, 76, 212, 215, 220, 223] [36, 37, 76, 77, 208, 213, 216, 221] [37, 38, 77, 78, 209, 214, 217, 222] [38, 39, 78, 79, 210, 215, 218, 223] [40, 47, 80, 87, 216, 219, 224, 227] [40, 41, 80, 81, 217, 220, 225, 228] [41, 42, 81, 82, 218, 221, 226, 229] [42, 43, 82, 83, 219, 222, 227, 230] [43, 44, 83, 84, 220, 223, 228, 231] [44, 45, 84, 85, 216, 221, 224, 229] [45, 46, 85, 86, 217, 222, 225, 230] [46, 47, 86, 87, 218, 223, 226, 231] [48, 55, 88, 95, 224, 227, 232, 235] [48, 49, 88, 89, 225, 228, 233, 236] [49, 50, 89, 90, 226, 229, 234, 237] [50, 51, 90, 91, 227, 230, 235, 238] [51, 52, 91, 92, 228, 231, 236, 239] [52, 53, 92, 93, 224, 229, 232, 237] [53, 54, 93, 94, 225, 230, 233, 238] [54, 55, 94, 95, 226, 231, 234, 239] [56, 63, 96, 103, 232, 235, 240, 243] [56, 57, 96, 97, 233, 236, 241, 244] [57, 58, 97, 98, 234, 237, 242, 245] [58, 59, 98, 99, 235, 238, 243, 246] [59, 60, 99, 100, 236, 239, 244, 247] [60, 61, 100, 101, 232, 237, 240, 245] [61, 62, 101, 102, 233, 238, 241, 246] [62, 63, 102, 103, 234, 239, 242, 247] [64, 71, 104, 111, 240, 243, 248, 251] [64, 65, 104, 105, 241, 244, 249, 252] [65, 66, 105, 106, 242, 245, 250, 253] [66, 67, 106, 107, 243, 246, 251, 254] [67, 68, 107, 108, 244, 247, 252, 255] [68, 69, 108, 109, 240, 245, 248, 253] [69, 70, 109, 110, 241, 246, 249, 254] [70, 71, 110, 111, 242, 247, 250, 255] [72, 79, 112, 119, 248, 251, 256, 259] [72, 73, 112, 113, 249, 252, 257, 260] [73, 74, 113, 114, 250, 253, 258, 261] [74, 75, 114, 115, 251, 254, 259, 262] [75, 76, 115, 116, 252, 255, 260, 263] [76, 77, 116, 117, 248, 253, 256, 261] [77, 78, 117, 118, 249, 254, 257, 262] [78, 79, 118, 119, 250, 255, 258, 263] [80, 87, 120, 127, 256, 259, 264, 267] [80, 81, 120, 121, 257, 260, 265, 268] [81, 82, 121, 122, 258, 261, 266, 269] [82, 83, 122, 123, 259, 262, 267, 270] [83, 84, 123, 124, 260, 263, 268, 271] [84, 85, 124, 125, 256, 261, 264, 269] [85, 86, 125, 126, 257, 262, 265, 270] [86, 87, 126, 127, 258, 263, 266, 271] [88, 95, 128, 135, 264, 267, 272, 275] [88, 89, 128, 129, 265, 268, 273, 276] [89, 90, 129, 130, 266, 269, 274, 277] [90, 91, 130, 131, 267, 270, 275, 278] [91, 92, 131, 132, 268, 271, 276, 279] [92, 93, 132, 133, 264, 269, 272, 277] [93, 94, 133, 134, 265, 270, 273, 278] [94, 95, 134, 135, 266, 271, 274, 279] [96, 103, 136, 143, 272, 275, 280, 283] [96, 97, 136, 137, 273, 276, 281, 284] [97, 98, 137, 138, 274, 277, 282, 285] [98, 99, 138, 139, 275, 278, 283, 286] [99, 100, 139, 140, 276, 279, 284, 287] [100, 101, 140, 141, 272, 277, 280, 285] [101, 102, 141, 142, 273, 278, 281, 286] [102, 103, 142, 143, 274, 279, 282, 287]
raw submission JSON