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[[390,82,31]] d ≤
n
390
k
82
d
31
kd²/n
202.056
w
16

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Distance

d_X 31 · witness weight 31 (claimed upper_bound)
witness operator (support, 31 qubits)
[9, 46, 61, 111, 118, 122, 136, 154, 163, 165, 168, 170, 188, 208, 210, 215, 224, 231, 236, 240, 252, 256, 263, 281, 311, 319, 324, 329, 349, 355, 374]
d_Z 31 · witness weight 31 (claimed upper_bound)
witness operator (support, 31 qubits)
[16, 35, 41, 61, 66, 71, 79, 109, 127, 134, 138, 150, 154, 159, 166, 175, 180, 182, 202, 220, 222, 225, 227, 236, 254, 268, 272, 279, 329, 344, 381]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @seunomonije
provenance submitted through the challenge
novelty novelty not audited
construction Generalized bicycle (two-block) code on the cyclic group Z_195: H_X = [A|B], H_Z = [B^T|A^T] with A, B circulants of a(x) = sum x^i, i in {0,2,7,16,55,93,111,147} (weight 8) and b(x) = sum x^i, i in {0,27,36,47,88,123,125,143} (weight 8); max check weight 16. Divisor trick: both a and b are weight-8 codewords of the classical cyclic code <g> for g = gcd(a, b, x195-1) of degree 41, giving k = 2*deg g = 82 by construction (machine-checked by rank). The weight-<=8 codeword pool of <g> saturates at 12 cyclic-shift classes (60k+ RIS trials, 3 seeds); this is pool-class pair (0,1), one of the 51 previously-unexplored pairs from the exhaustive 66-pair enumeration of that pool. Distance evidence: randomized-information-set screen read d<=43 at 12k trials; staged deep ladder with fresh seed per rung (pair_depth 8) read 35 at 2M, 34 at 8M, 32 at 24M trials; final five-config barrage read [31, 32, 33, 36, 35] = three 8M-trial passes at pair_depth 8 with fresh seeds 777/999/424242 plus pair_depth-16 and pair_depth-24 passes at 3M trials. The lightest logical ever witnessed is the weight-31 Z logical from the seed-777 pass (support embedded below); the GB transpose symmetry (block swap + per-block index reversal i -> -i mod 195) maps it to the embedded weight-31 X logical, machine-verified against ker/rowspace. Claimed d <= 31 = the lightest witnessed logical.
model Claude Claude Fable 5 (claimed, not verified)
builds on arXiv:1904.02703, arXiv:2203.17216
date 2026-07-18
notes Search and verification by Claude Fable 5. Distance is an honest upper bound from ~30M randomized-information-set trials across 8 independent seeded passes and three pair depths; no logical below weight 31 was ever witnessed, but d could be lower. Literature novelty unverified. kd^2/n = 202.06; supersedes the [[390,82,30]] entry from the same divisor pool.
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 195 · Z-checks 195
H_X (195 checks, sparse supports)
[0, 2, 7, 16, 55, 93, 111, 147, 195, 222, 231, 242, 283, 318, 320, 338] [1, 3, 8, 17, 56, 94, 112, 148, 196, 223, 232, 243, 284, 319, 321, 339] [2, 4, 9, 18, 57, 95, 113, 149, 197, 224, 233, 244, 285, 320, 322, 340] [3, 5, 10, 19, 58, 96, 114, 150, 198, 225, 234, 245, 286, 321, 323, 341] [4, 6, 11, 20, 59, 97, 115, 151, 199, 226, 235, 246, 287, 322, 324, 342] [5, 7, 12, 21, 60, 98, 116, 152, 200, 227, 236, 247, 288, 323, 325, 343] [6, 8, 13, 22, 61, 99, 117, 153, 201, 228, 237, 248, 289, 324, 326, 344] [7, 9, 14, 23, 62, 100, 118, 154, 202, 229, 238, 249, 290, 325, 327, 345] [8, 10, 15, 24, 63, 101, 119, 155, 203, 230, 239, 250, 291, 326, 328, 346] [9, 11, 16, 25, 64, 102, 120, 156, 204, 231, 240, 251, 292, 327, 329, 347] [10, 12, 17, 26, 65, 103, 121, 157, 205, 232, 241, 252, 293, 328, 330, 348] [11, 13, 18, 27, 66, 104, 122, 158, 206, 233, 242, 253, 294, 329, 331, 349] [12, 14, 19, 28, 67, 105, 123, 159, 207, 234, 243, 254, 295, 330, 332, 350] [13, 15, 20, 29, 68, 106, 124, 160, 208, 235, 244, 255, 296, 331, 333, 351] [14, 16, 21, 30, 69, 107, 125, 161, 209, 236, 245, 256, 297, 332, 334, 352] [15, 17, 22, 31, 70, 108, 126, 162, 210, 237, 246, 257, 298, 333, 335, 353] [16, 18, 23, 32, 71, 109, 127, 163, 211, 238, 247, 258, 299, 334, 336, 354] [17, 19, 24, 33, 72, 110, 128, 164, 212, 239, 248, 259, 300, 335, 337, 355] [18, 20, 25, 34, 73, 111, 129, 165, 213, 240, 249, 260, 301, 336, 338, 356] [19, 21, 26, 35, 74, 112, 130, 166, 214, 241, 250, 261, 302, 337, 339, 357] [20, 22, 27, 36, 75, 113, 131, 167, 215, 242, 251, 262, 303, 338, 340, 358] [21, 23, 28, 37, 76, 114, 132, 168, 216, 243, 252, 263, 304, 339, 341, 359] [22, 24, 29, 38, 77, 115, 133, 169, 217, 244, 253, 264, 305, 340, 342, 360] [23, 25, 30, 39, 78, 116, 134, 170, 218, 245, 254, 265, 306, 341, 343, 361] [24, 26, 31, 40, 79, 117, 135, 171, 219, 246, 255, 266, 307, 342, 344, 362] [25, 27, 32, 41, 80, 118, 136, 172, 220, 247, 256, 267, 308, 343, 345, 363] [26, 28, 33, 42, 81, 119, 137, 173, 221, 248, 257, 268, 309, 344, 346, 364] [27, 29, 34, 43, 82, 120, 138, 174, 222, 249, 258, 269, 310, 345, 347, 365] [28, 30, 35, 44, 83, 121, 139, 175, 223, 250, 259, 270, 311, 346, 348, 366] [29, 31, 36, 45, 84, 122, 140, 176, 224, 251, 260, 271, 312, 347, 349, 367] [30, 32, 37, 46, 85, 123, 141, 177, 225, 252, 261, 272, 313, 348, 350, 368] [31, 33, 38, 47, 86, 124, 142, 178, 226, 253, 262, 273, 314, 349, 351, 369] [32, 34, 39, 48, 87, 125, 143, 179, 227, 254, 263, 274, 315, 350, 352, 370] [33, 35, 40, 49, 88, 126, 144, 180, 228, 255, 264, 275, 316, 351, 353, 371] [34, 36, 41, 50, 89, 127, 145, 181, 229, 256, 265, 276, 317, 352, 354, 372] [35, 37, 42, 51, 90, 128, 146, 182, 230, 257, 266, 277, 318, 353, 355, 373] [36, 38, 43, 52, 91, 129, 147, 183, 231, 258, 267, 278, 319, 354, 356, 374] [37, 39, 44, 53, 92, 130, 148, 184, 232, 259, 268, 279, 320, 355, 357, 375] [38, 40, 45, 54, 93, 131, 149, 185, 233, 260, 269, 280, 321, 356, 358, 376] [39, 41, 46, 55, 94, 132, 150, 186, 234, 261, 270, 281, 322, 357, 359, 377] [40, 42, 47, 56, 95, 133, 151, 187, 235, 262, 271, 282, 323, 358, 360, 378] [41, 43, 48, 57, 96, 134, 152, 188, 236, 263, 272, 283, 324, 359, 361, 379] [42, 44, 49, 58, 97, 135, 153, 189, 237, 264, 273, 284, 325, 360, 362, 380] [43, 45, 50, 59, 98, 136, 154, 190, 238, 265, 274, 285, 326, 361, 363, 381] [44, 46, 51, 60, 99, 137, 155, 191, 239, 266, 275, 286, 327, 362, 364, 382] [45, 47, 52, 61, 100, 138, 156, 192, 240, 267, 276, 287, 328, 363, 365, 383] [46, 48, 53, 62, 101, 139, 157, 193, 241, 268, 277, 288, 329, 364, 366, 384] [47, 49, 54, 63, 102, 140, 158, 194, 242, 269, 278, 289, 330, 365, 367, 385] [0, 48, 50, 55, 64, 103, 141, 159, 243, 270, 279, 290, 331, 366, 368, 386] [1, 49, 51, 56, 65, 104, 142, 160, 244, 271, 280, 291, 332, 367, 369, 387] [2, 50, 52, 57, 66, 105, 143, 161, 245, 272, 281, 292, 333, 368, 370, 388] [3, 51, 53, 58, 67, 106, 144, 162, 246, 273, 282, 293, 334, 369, 371, 389] [4, 52, 54, 59, 68, 107, 145, 163, 195, 247, 274, 283, 294, 335, 370, 372] [5, 53, 55, 60, 69, 108, 146, 164, 196, 248, 275, 284, 295, 336, 371, 373] [6, 54, 56, 61, 70, 109, 147, 165, 197, 249, 276, 285, 296, 337, 372, 374] [7, 55, 57, 62, 71, 110, 148, 166, 198, 250, 277, 286, 297, 338, 373, 375] [8, 56, 58, 63, 72, 111, 149, 167, 199, 251, 278, 287, 298, 339, 374, 376] [9, 57, 59, 64, 73, 112, 150, 168, 200, 252, 279, 288, 299, 340, 375, 377] [10, 58, 60, 65, 74, 113, 151, 169, 201, 253, 280, 289, 300, 341, 376, 378] [11, 59, 61, 66, 75, 114, 152, 170, 202, 254, 281, 290, 301, 342, 377, 379] [12, 60, 62, 67, 76, 115, 153, 171, 203, 255, 282, 291, 302, 343, 378, 380] [13, 61, 63, 68, 77, 116, 154, 172, 204, 256, 283, 292, 303, 344, 379, 381] [14, 62, 64, 69, 78, 117, 155, 173, 205, 257, 284, 293, 304, 345, 380, 382] [15, 63, 65, 70, 79, 118, 156, 174, 206, 258, 285, 294, 305, 346, 381, 383] [16, 64, 66, 71, 80, 119, 157, 175, 207, 259, 286, 295, 306, 347, 382, 384] [17, 65, 67, 72, 81, 120, 158, 176, 208, 260, 287, 296, 307, 348, 383, 385] [18, 66, 68, 73, 82, 121, 159, 177, 209, 261, 288, 297, 308, 349, 384, 386] [19, 67, 69, 74, 83, 122, 160, 178, 210, 262, 289, 298, 309, 350, 385, 387] [20, 68, 70, 75, 84, 123, 161, 179, 211, 263, 290, 299, 310, 351, 386, 388] [21, 69, 71, 76, 85, 124, 162, 180, 212, 264, 291, 300, 311, 352, 387, 389] [22, 70, 72, 77, 86, 125, 163, 181, 195, 213, 265, 292, 301, 312, 353, 388] [23, 71, 73, 78, 87, 126, 164, 182, 196, 214, 266, 293, 302, 313, 354, 389] [24, 72, 74, 79, 88, 127, 165, 183, 195, 197, 215, 267, 294, 303, 314, 355] [25, 73, 75, 80, 89, 128, 166, 184, 196, 198, 216, 268, 295, 304, 315, 356] [26, 74, 76, 81, 90, 129, 167, 185, 197, 199, 217, 269, 296, 305, 316, 357] [27, 75, 77, 82, 91, 130, 168, 186, 198, 200, 218, 270, 297, 306, 317, 358] [28, 76, 78, 83, 92, 131, 169, 187, 199, 201, 219, 271, 298, 307, 318, 359] [29, 77, 79, 84, 93, 132, 170, 188, 200, 202, 220, 272, 299, 308, 319, 360] [30, 78, 80, 85, 94, 133, 171, 189, 201, 203, 221, 273, 300, 309, 320, 361] [31, 79, 81, 86, 95, 134, 172, 190, 202, 204, 222, 274, 301, 310, 321, 362] [32, 80, 82, 87, 96, 135, 173, 191, 203, 205, 223, 275, 302, 311, 322, 363] [33, 81, 83, 88, 97, 136, 174, 192, 204, 206, 224, 276, 303, 312, 323, 364] [34, 82, 84, 89, 98, 137, 175, 193, 205, 207, 225, 277, 304, 313, 324, 365] [35, 83, 85, 90, 99, 138, 176, 194, 206, 208, 226, 278, 305, 314, 325, 366] [0, 36, 84, 86, 91, 100, 139, 177, 207, 209, 227, 279, 306, 315, 326, 367] [1, 37, 85, 87, 92, 101, 140, 178, 208, 210, 228, 280, 307, 316, 327, 368] [2, 38, 86, 88, 93, 102, 141, 179, 209, 211, 229, 281, 308, 317, 328, 369] [3, 39, 87, 89, 94, 103, 142, 180, 210, 212, 230, 282, 309, 318, 329, 370] [4, 40, 88, 90, 95, 104, 143, 181, 211, 213, 231, 283, 310, 319, 330, 371] [5, 41, 89, 91, 96, 105, 144, 182, 212, 214, 232, 284, 311, 320, 331, 372] [6, 42, 90, 92, 97, 106, 145, 183, 213, 215, 233, 285, 312, 321, 332, 373] [7, 43, 91, 93, 98, 107, 146, 184, 214, 216, 234, 286, 313, 322, 333, 374] [8, 44, 92, 94, 99, 108, 147, 185, 215, 217, 235, 287, 314, 323, 334, 375] [9, 45, 93, 95, 100, 109, 148, 186, 216, 218, 236, 288, 315, 324, 335, 376] [10, 46, 94, 96, 101, 110, 149, 187, 217, 219, 237, 289, 316, 325, 336, 377] [11, 47, 95, 97, 102, 111, 150, 188, 218, 220, 238, 290, 317, 326, 337, 378] [12, 48, 96, 98, 103, 112, 151, 189, 219, 221, 239, 291, 318, 327, 338, 379] [13, 49, 97, 99, 104, 113, 152, 190, 220, 222, 240, 292, 319, 328, 339, 380] [14, 50, 98, 100, 105, 114, 153, 191, 221, 223, 241, 293, 320, 329, 340, 381] [15, 51, 99, 101, 106, 115, 154, 192, 222, 224, 242, 294, 321, 330, 341, 382] [16, 52, 100, 102, 107, 116, 155, 193, 223, 225, 243, 295, 322, 331, 342, 383] [17, 53, 101, 103, 108, 117, 156, 194, 224, 226, 244, 296, 323, 332, 343, 384] [0, 18, 54, 102, 104, 109, 118, 157, 225, 227, 245, 297, 324, 333, 344, 385] [1, 19, 55, 103, 105, 110, 119, 158, 226, 228, 246, 298, 325, 334, 345, 386] [2, 20, 56, 104, 106, 111, 120, 159, 227, 229, 247, 299, 326, 335, 346, 387] [3, 21, 57, 105, 107, 112, 121, 160, 228, 230, 248, 300, 327, 336, 347, 388] [4, 22, 58, 106, 108, 113, 122, 161, 229, 231, 249, 301, 328, 337, 348, 389] [5, 23, 59, 107, 109, 114, 123, 162, 195, 230, 232, 250, 302, 329, 338, 349] [6, 24, 60, 108, 110, 115, 124, 163, 196, 231, 233, 251, 303, 330, 339, 350] [7, 25, 61, 109, 111, 116, 125, 164, 197, 232, 234, 252, 304, 331, 340, 351] [8, 26, 62, 110, 112, 117, 126, 165, 198, 233, 235, 253, 305, 332, 341, 352] [9, 27, 63, 111, 113, 118, 127, 166, 199, 234, 236, 254, 306, 333, 342, 353] [10, 28, 64, 112, 114, 119, 128, 167, 200, 235, 237, 255, 307, 334, 343, 354] [11, 29, 65, 113, 115, 120, 129, 168, 201, 236, 238, 256, 308, 335, 344, 355] [12, 30, 66, 114, 116, 121, 130, 169, 202, 237, 239, 257, 309, 336, 345, 356] [13, 31, 67, 115, 117, 122, 131, 170, 203, 238, 240, 258, 310, 337, 346, 357] [14, 32, 68, 116, 118, 123, 132, 171, 204, 239, 241, 259, 311, 338, 347, 358] [15, 33, 69, 117, 119, 124, 133, 172, 205, 240, 242, 260, 312, 339, 348, 359] [16, 34, 70, 118, 120, 125, 134, 173, 206, 241, 243, 261, 313, 340, 349, 360] [17, 35, 71, 119, 121, 126, 135, 174, 207, 242, 244, 262, 314, 341, 350, 361] [18, 36, 72, 120, 122, 127, 136, 175, 208, 243, 245, 263, 315, 342, 351, 362] [19, 37, 73, 121, 123, 128, 137, 176, 209, 244, 246, 264, 316, 343, 352, 363] [20, 38, 74, 122, 124, 129, 138, 177, 210, 245, 247, 265, 317, 344, 353, 364] [21, 39, 75, 123, 125, 130, 139, 178, 211, 246, 248, 266, 318, 345, 354, 365] [22, 40, 76, 124, 126, 131, 140, 179, 212, 247, 249, 267, 319, 346, 355, 366] [23, 41, 77, 125, 127, 132, 141, 180, 213, 248, 250, 268, 320, 347, 356, 367] [24, 42, 78, 126, 128, 133, 142, 181, 214, 249, 251, 269, 321, 348, 357, 368] [25, 43, 79, 127, 129, 134, 143, 182, 215, 250, 252, 270, 322, 349, 358, 369] [26, 44, 80, 128, 130, 135, 144, 183, 216, 251, 253, 271, 323, 350, 359, 370] [27, 45, 81, 129, 131, 136, 145, 184, 217, 252, 254, 272, 324, 351, 360, 371] [28, 46, 82, 130, 132, 137, 146, 185, 218, 253, 255, 273, 325, 352, 361, 372] [29, 47, 83, 131, 133, 138, 147, 186, 219, 254, 256, 274, 326, 353, 362, 373] [30, 48, 84, 132, 134, 139, 148, 187, 220, 255, 257, 275, 327, 354, 363, 374] [31, 49, 85, 133, 135, 140, 149, 188, 221, 256, 258, 276, 328, 355, 364, 375] [32, 50, 86, 134, 136, 141, 150, 189, 222, 257, 259, 277, 329, 356, 365, 376] [33, 51, 87, 135, 137, 142, 151, 190, 223, 258, 260, 278, 330, 357, 366, 377] [34, 52, 88, 136, 138, 143, 152, 191, 224, 259, 261, 279, 331, 358, 367, 378] [35, 53, 89, 137, 139, 144, 153, 192, 225, 260, 262, 280, 332, 359, 368, 379] [36, 54, 90, 138, 140, 145, 154, 193, 226, 261, 263, 281, 333, 360, 369, 380] [37, 55, 91, 139, 141, 146, 155, 194, 227, 262, 264, 282, 334, 361, 370, 381] [0, 38, 56, 92, 140, 142, 147, 156, 228, 263, 265, 283, 335, 362, 371, 382] [1, 39, 57, 93, 141, 143, 148, 157, 229, 264, 266, 284, 336, 363, 372, 383] [2, 40, 58, 94, 142, 144, 149, 158, 230, 265, 267, 285, 337, 364, 373, 384] [3, 41, 59, 95, 143, 145, 150, 159, 231, 266, 268, 286, 338, 365, 374, 385] [4, 42, 60, 96, 144, 146, 151, 160, 232, 267, 269, 287, 339, 366, 375, 386] [5, 43, 61, 97, 145, 147, 152, 161, 233, 268, 270, 288, 340, 367, 376, 387] [6, 44, 62, 98, 146, 148, 153, 162, 234, 269, 271, 289, 341, 368, 377, 388] [7, 45, 63, 99, 147, 149, 154, 163, 235, 270, 272, 290, 342, 369, 378, 389] [8, 46, 64, 100, 148, 150, 155, 164, 195, 236, 271, 273, 291, 343, 370, 379] [9, 47, 65, 101, 149, 151, 156, 165, 196, 237, 272, 274, 292, 344, 371, 380] [10, 48, 66, 102, 150, 152, 157, 166, 197, 238, 273, 275, 293, 345, 372, 381] [11, 49, 67, 103, 151, 153, 158, 167, 198, 239, 274, 276, 294, 346, 373, 382] [12, 50, 68, 104, 152, 154, 159, 168, 199, 240, 275, 277, 295, 347, 374, 383] [13, 51, 69, 105, 153, 155, 160, 169, 200, 241, 276, 278, 296, 348, 375, 384] [14, 52, 70, 106, 154, 156, 161, 170, 201, 242, 277, 279, 297, 349, 376, 385] [15, 53, 71, 107, 155, 157, 162, 171, 202, 243, 278, 280, 298, 350, 377, 386] [16, 54, 72, 108, 156, 158, 163, 172, 203, 244, 279, 281, 299, 351, 378, 387] [17, 55, 73, 109, 157, 159, 164, 173, 204, 245, 280, 282, 300, 352, 379, 388] [18, 56, 74, 110, 158, 160, 165, 174, 205, 246, 281, 283, 301, 353, 380, 389] [19, 57, 75, 111, 159, 161, 166, 175, 195, 206, 247, 282, 284, 302, 354, 381] [20, 58, 76, 112, 160, 162, 167, 176, 196, 207, 248, 283, 285, 303, 355, 382] [21, 59, 77, 113, 161, 163, 168, 177, 197, 208, 249, 284, 286, 304, 356, 383] [22, 60, 78, 114, 162, 164, 169, 178, 198, 209, 250, 285, 287, 305, 357, 384] [23, 61, 79, 115, 163, 165, 170, 179, 199, 210, 251, 286, 288, 306, 358, 385] [24, 62, 80, 116, 164, 166, 171, 180, 200, 211, 252, 287, 289, 307, 359, 386] [25, 63, 81, 117, 165, 167, 172, 181, 201, 212, 253, 288, 290, 308, 360, 387] [26, 64, 82, 118, 166, 168, 173, 182, 202, 213, 254, 289, 291, 309, 361, 388] [27, 65, 83, 119, 167, 169, 174, 183, 203, 214, 255, 290, 292, 310, 362, 389] [28, 66, 84, 120, 168, 170, 175, 184, 195, 204, 215, 256, 291, 293, 311, 363] [29, 67, 85, 121, 169, 171, 176, 185, 196, 205, 216, 257, 292, 294, 312, 364] [30, 68, 86, 122, 170, 172, 177, 186, 197, 206, 217, 258, 293, 295, 313, 365] [31, 69, 87, 123, 171, 173, 178, 187, 198, 207, 218, 259, 294, 296, 314, 366] [32, 70, 88, 124, 172, 174, 179, 188, 199, 208, 219, 260, 295, 297, 315, 367] [33, 71, 89, 125, 173, 175, 180, 189, 200, 209, 220, 261, 296, 298, 316, 368] [34, 72, 90, 126, 174, 176, 181, 190, 201, 210, 221, 262, 297, 299, 317, 369] [35, 73, 91, 127, 175, 177, 182, 191, 202, 211, 222, 263, 298, 300, 318, 370] [36, 74, 92, 128, 176, 178, 183, 192, 203, 212, 223, 264, 299, 301, 319, 371] [37, 75, 93, 129, 177, 179, 184, 193, 204, 213, 224, 265, 300, 302, 320, 372] [38, 76, 94, 130, 178, 180, 185, 194, 205, 214, 225, 266, 301, 303, 321, 373] [0, 39, 77, 95, 131, 179, 181, 186, 206, 215, 226, 267, 302, 304, 322, 374] [1, 40, 78, 96, 132, 180, 182, 187, 207, 216, 227, 268, 303, 305, 323, 375] [2, 41, 79, 97, 133, 181, 183, 188, 208, 217, 228, 269, 304, 306, 324, 376] [3, 42, 80, 98, 134, 182, 184, 189, 209, 218, 229, 270, 305, 307, 325, 377] [4, 43, 81, 99, 135, 183, 185, 190, 210, 219, 230, 271, 306, 308, 326, 378] [5, 44, 82, 100, 136, 184, 186, 191, 211, 220, 231, 272, 307, 309, 327, 379] [6, 45, 83, 101, 137, 185, 187, 192, 212, 221, 232, 273, 308, 310, 328, 380] [7, 46, 84, 102, 138, 186, 188, 193, 213, 222, 233, 274, 309, 311, 329, 381] [8, 47, 85, 103, 139, 187, 189, 194, 214, 223, 234, 275, 310, 312, 330, 382] [0, 9, 48, 86, 104, 140, 188, 190, 215, 224, 235, 276, 311, 313, 331, 383] [1, 10, 49, 87, 105, 141, 189, 191, 216, 225, 236, 277, 312, 314, 332, 384] [2, 11, 50, 88, 106, 142, 190, 192, 217, 226, 237, 278, 313, 315, 333, 385] [3, 12, 51, 89, 107, 143, 191, 193, 218, 227, 238, 279, 314, 316, 334, 386] [4, 13, 52, 90, 108, 144, 192, 194, 219, 228, 239, 280, 315, 317, 335, 387] [0, 5, 14, 53, 91, 109, 145, 193, 220, 229, 240, 281, 316, 318, 336, 388] [1, 6, 15, 54, 92, 110, 146, 194, 221, 230, 241, 282, 317, 319, 337, 389]
H_Z (195 checks, sparse supports)
[0, 52, 70, 72, 107, 148, 159, 168, 195, 243, 279, 297, 335, 374, 383, 388] [1, 53, 71, 73, 108, 149, 160, 169, 196, 244, 280, 298, 336, 375, 384, 389] [2, 54, 72, 74, 109, 150, 161, 170, 195, 197, 245, 281, 299, 337, 376, 385] [3, 55, 73, 75, 110, 151, 162, 171, 196, 198, 246, 282, 300, 338, 377, 386] [4, 56, 74, 76, 111, 152, 163, 172, 197, 199, 247, 283, 301, 339, 378, 387] [5, 57, 75, 77, 112, 153, 164, 173, 198, 200, 248, 284, 302, 340, 379, 388] [6, 58, 76, 78, 113, 154, 165, 174, 199, 201, 249, 285, 303, 341, 380, 389] [7, 59, 77, 79, 114, 155, 166, 175, 195, 200, 202, 250, 286, 304, 342, 381] [8, 60, 78, 80, 115, 156, 167, 176, 196, 201, 203, 251, 287, 305, 343, 382] [9, 61, 79, 81, 116, 157, 168, 177, 197, 202, 204, 252, 288, 306, 344, 383] [10, 62, 80, 82, 117, 158, 169, 178, 198, 203, 205, 253, 289, 307, 345, 384] [11, 63, 81, 83, 118, 159, 170, 179, 199, 204, 206, 254, 290, 308, 346, 385] [12, 64, 82, 84, 119, 160, 171, 180, 200, 205, 207, 255, 291, 309, 347, 386] [13, 65, 83, 85, 120, 161, 172, 181, 201, 206, 208, 256, 292, 310, 348, 387] [14, 66, 84, 86, 121, 162, 173, 182, 202, 207, 209, 257, 293, 311, 349, 388] [15, 67, 85, 87, 122, 163, 174, 183, 203, 208, 210, 258, 294, 312, 350, 389] [16, 68, 86, 88, 123, 164, 175, 184, 195, 204, 209, 211, 259, 295, 313, 351] [17, 69, 87, 89, 124, 165, 176, 185, 196, 205, 210, 212, 260, 296, 314, 352] [18, 70, 88, 90, 125, 166, 177, 186, 197, 206, 211, 213, 261, 297, 315, 353] [19, 71, 89, 91, 126, 167, 178, 187, 198, 207, 212, 214, 262, 298, 316, 354] [20, 72, 90, 92, 127, 168, 179, 188, 199, 208, 213, 215, 263, 299, 317, 355] [21, 73, 91, 93, 128, 169, 180, 189, 200, 209, 214, 216, 264, 300, 318, 356] [22, 74, 92, 94, 129, 170, 181, 190, 201, 210, 215, 217, 265, 301, 319, 357] [23, 75, 93, 95, 130, 171, 182, 191, 202, 211, 216, 218, 266, 302, 320, 358] [24, 76, 94, 96, 131, 172, 183, 192, 203, 212, 217, 219, 267, 303, 321, 359] [25, 77, 95, 97, 132, 173, 184, 193, 204, 213, 218, 220, 268, 304, 322, 360] [26, 78, 96, 98, 133, 174, 185, 194, 205, 214, 219, 221, 269, 305, 323, 361] [0, 27, 79, 97, 99, 134, 175, 186, 206, 215, 220, 222, 270, 306, 324, 362] [1, 28, 80, 98, 100, 135, 176, 187, 207, 216, 221, 223, 271, 307, 325, 363] [2, 29, 81, 99, 101, 136, 177, 188, 208, 217, 222, 224, 272, 308, 326, 364] [3, 30, 82, 100, 102, 137, 178, 189, 209, 218, 223, 225, 273, 309, 327, 365] [4, 31, 83, 101, 103, 138, 179, 190, 210, 219, 224, 226, 274, 310, 328, 366] [5, 32, 84, 102, 104, 139, 180, 191, 211, 220, 225, 227, 275, 311, 329, 367] [6, 33, 85, 103, 105, 140, 181, 192, 212, 221, 226, 228, 276, 312, 330, 368] [7, 34, 86, 104, 106, 141, 182, 193, 213, 222, 227, 229, 277, 313, 331, 369] [8, 35, 87, 105, 107, 142, 183, 194, 214, 223, 228, 230, 278, 314, 332, 370] [0, 9, 36, 88, 106, 108, 143, 184, 215, 224, 229, 231, 279, 315, 333, 371] [1, 10, 37, 89, 107, 109, 144, 185, 216, 225, 230, 232, 280, 316, 334, 372] [2, 11, 38, 90, 108, 110, 145, 186, 217, 226, 231, 233, 281, 317, 335, 373] [3, 12, 39, 91, 109, 111, 146, 187, 218, 227, 232, 234, 282, 318, 336, 374] [4, 13, 40, 92, 110, 112, 147, 188, 219, 228, 233, 235, 283, 319, 337, 375] [5, 14, 41, 93, 111, 113, 148, 189, 220, 229, 234, 236, 284, 320, 338, 376] [6, 15, 42, 94, 112, 114, 149, 190, 221, 230, 235, 237, 285, 321, 339, 377] [7, 16, 43, 95, 113, 115, 150, 191, 222, 231, 236, 238, 286, 322, 340, 378] [8, 17, 44, 96, 114, 116, 151, 192, 223, 232, 237, 239, 287, 323, 341, 379] [9, 18, 45, 97, 115, 117, 152, 193, 224, 233, 238, 240, 288, 324, 342, 380] [10, 19, 46, 98, 116, 118, 153, 194, 225, 234, 239, 241, 289, 325, 343, 381] [0, 11, 20, 47, 99, 117, 119, 154, 226, 235, 240, 242, 290, 326, 344, 382] [1, 12, 21, 48, 100, 118, 120, 155, 227, 236, 241, 243, 291, 327, 345, 383] [2, 13, 22, 49, 101, 119, 121, 156, 228, 237, 242, 244, 292, 328, 346, 384] [3, 14, 23, 50, 102, 120, 122, 157, 229, 238, 243, 245, 293, 329, 347, 385] [4, 15, 24, 51, 103, 121, 123, 158, 230, 239, 244, 246, 294, 330, 348, 386] [5, 16, 25, 52, 104, 122, 124, 159, 231, 240, 245, 247, 295, 331, 349, 387] [6, 17, 26, 53, 105, 123, 125, 160, 232, 241, 246, 248, 296, 332, 350, 388] [7, 18, 27, 54, 106, 124, 126, 161, 233, 242, 247, 249, 297, 333, 351, 389] [8, 19, 28, 55, 107, 125, 127, 162, 195, 234, 243, 248, 250, 298, 334, 352] [9, 20, 29, 56, 108, 126, 128, 163, 196, 235, 244, 249, 251, 299, 335, 353] [10, 21, 30, 57, 109, 127, 129, 164, 197, 236, 245, 250, 252, 300, 336, 354] [11, 22, 31, 58, 110, 128, 130, 165, 198, 237, 246, 251, 253, 301, 337, 355] [12, 23, 32, 59, 111, 129, 131, 166, 199, 238, 247, 252, 254, 302, 338, 356] [13, 24, 33, 60, 112, 130, 132, 167, 200, 239, 248, 253, 255, 303, 339, 357] [14, 25, 34, 61, 113, 131, 133, 168, 201, 240, 249, 254, 256, 304, 340, 358] [15, 26, 35, 62, 114, 132, 134, 169, 202, 241, 250, 255, 257, 305, 341, 359] [16, 27, 36, 63, 115, 133, 135, 170, 203, 242, 251, 256, 258, 306, 342, 360] [17, 28, 37, 64, 116, 134, 136, 171, 204, 243, 252, 257, 259, 307, 343, 361] [18, 29, 38, 65, 117, 135, 137, 172, 205, 244, 253, 258, 260, 308, 344, 362] [19, 30, 39, 66, 118, 136, 138, 173, 206, 245, 254, 259, 261, 309, 345, 363] [20, 31, 40, 67, 119, 137, 139, 174, 207, 246, 255, 260, 262, 310, 346, 364] [21, 32, 41, 68, 120, 138, 140, 175, 208, 247, 256, 261, 263, 311, 347, 365] [22, 33, 42, 69, 121, 139, 141, 176, 209, 248, 257, 262, 264, 312, 348, 366] [23, 34, 43, 70, 122, 140, 142, 177, 210, 249, 258, 263, 265, 313, 349, 367] [24, 35, 44, 71, 123, 141, 143, 178, 211, 250, 259, 264, 266, 314, 350, 368] [25, 36, 45, 72, 124, 142, 144, 179, 212, 251, 260, 265, 267, 315, 351, 369] [26, 37, 46, 73, 125, 143, 145, 180, 213, 252, 261, 266, 268, 316, 352, 370] [27, 38, 47, 74, 126, 144, 146, 181, 214, 253, 262, 267, 269, 317, 353, 371] [28, 39, 48, 75, 127, 145, 147, 182, 215, 254, 263, 268, 270, 318, 354, 372] [29, 40, 49, 76, 128, 146, 148, 183, 216, 255, 264, 269, 271, 319, 355, 373] [30, 41, 50, 77, 129, 147, 149, 184, 217, 256, 265, 270, 272, 320, 356, 374] [31, 42, 51, 78, 130, 148, 150, 185, 218, 257, 266, 271, 273, 321, 357, 375] [32, 43, 52, 79, 131, 149, 151, 186, 219, 258, 267, 272, 274, 322, 358, 376] [33, 44, 53, 80, 132, 150, 152, 187, 220, 259, 268, 273, 275, 323, 359, 377] [34, 45, 54, 81, 133, 151, 153, 188, 221, 260, 269, 274, 276, 324, 360, 378] [35, 46, 55, 82, 134, 152, 154, 189, 222, 261, 270, 275, 277, 325, 361, 379] [36, 47, 56, 83, 135, 153, 155, 190, 223, 262, 271, 276, 278, 326, 362, 380] [37, 48, 57, 84, 136, 154, 156, 191, 224, 263, 272, 277, 279, 327, 363, 381] [38, 49, 58, 85, 137, 155, 157, 192, 225, 264, 273, 278, 280, 328, 364, 382] [39, 50, 59, 86, 138, 156, 158, 193, 226, 265, 274, 279, 281, 329, 365, 383] [40, 51, 60, 87, 139, 157, 159, 194, 227, 266, 275, 280, 282, 330, 366, 384] [0, 41, 52, 61, 88, 140, 158, 160, 228, 267, 276, 281, 283, 331, 367, 385] [1, 42, 53, 62, 89, 141, 159, 161, 229, 268, 277, 282, 284, 332, 368, 386] [2, 43, 54, 63, 90, 142, 160, 162, 230, 269, 278, 283, 285, 333, 369, 387] [3, 44, 55, 64, 91, 143, 161, 163, 231, 270, 279, 284, 286, 334, 370, 388] [4, 45, 56, 65, 92, 144, 162, 164, 232, 271, 280, 285, 287, 335, 371, 389] [5, 46, 57, 66, 93, 145, 163, 165, 195, 233, 272, 281, 286, 288, 336, 372] [6, 47, 58, 67, 94, 146, 164, 166, 196, 234, 273, 282, 287, 289, 337, 373] [7, 48, 59, 68, 95, 147, 165, 167, 197, 235, 274, 283, 288, 290, 338, 374] [8, 49, 60, 69, 96, 148, 166, 168, 198, 236, 275, 284, 289, 291, 339, 375] [9, 50, 61, 70, 97, 149, 167, 169, 199, 237, 276, 285, 290, 292, 340, 376] [10, 51, 62, 71, 98, 150, 168, 170, 200, 238, 277, 286, 291, 293, 341, 377] [11, 52, 63, 72, 99, 151, 169, 171, 201, 239, 278, 287, 292, 294, 342, 378] [12, 53, 64, 73, 100, 152, 170, 172, 202, 240, 279, 288, 293, 295, 343, 379] [13, 54, 65, 74, 101, 153, 171, 173, 203, 241, 280, 289, 294, 296, 344, 380] [14, 55, 66, 75, 102, 154, 172, 174, 204, 242, 281, 290, 295, 297, 345, 381] [15, 56, 67, 76, 103, 155, 173, 175, 205, 243, 282, 291, 296, 298, 346, 382] [16, 57, 68, 77, 104, 156, 174, 176, 206, 244, 283, 292, 297, 299, 347, 383] [17, 58, 69, 78, 105, 157, 175, 177, 207, 245, 284, 293, 298, 300, 348, 384] [18, 59, 70, 79, 106, 158, 176, 178, 208, 246, 285, 294, 299, 301, 349, 385] [19, 60, 71, 80, 107, 159, 177, 179, 209, 247, 286, 295, 300, 302, 350, 386] [20, 61, 72, 81, 108, 160, 178, 180, 210, 248, 287, 296, 301, 303, 351, 387] [21, 62, 73, 82, 109, 161, 179, 181, 211, 249, 288, 297, 302, 304, 352, 388] [22, 63, 74, 83, 110, 162, 180, 182, 212, 250, 289, 298, 303, 305, 353, 389] [23, 64, 75, 84, 111, 163, 181, 183, 195, 213, 251, 290, 299, 304, 306, 354] [24, 65, 76, 85, 112, 164, 182, 184, 196, 214, 252, 291, 300, 305, 307, 355] [25, 66, 77, 86, 113, 165, 183, 185, 197, 215, 253, 292, 301, 306, 308, 356] [26, 67, 78, 87, 114, 166, 184, 186, 198, 216, 254, 293, 302, 307, 309, 357] [27, 68, 79, 88, 115, 167, 185, 187, 199, 217, 255, 294, 303, 308, 310, 358] [28, 69, 80, 89, 116, 168, 186, 188, 200, 218, 256, 295, 304, 309, 311, 359] [29, 70, 81, 90, 117, 169, 187, 189, 201, 219, 257, 296, 305, 310, 312, 360] [30, 71, 82, 91, 118, 170, 188, 190, 202, 220, 258, 297, 306, 311, 313, 361] [31, 72, 83, 92, 119, 171, 189, 191, 203, 221, 259, 298, 307, 312, 314, 362] [32, 73, 84, 93, 120, 172, 190, 192, 204, 222, 260, 299, 308, 313, 315, 363] [33, 74, 85, 94, 121, 173, 191, 193, 205, 223, 261, 300, 309, 314, 316, 364] [34, 75, 86, 95, 122, 174, 192, 194, 206, 224, 262, 301, 310, 315, 317, 365] [0, 35, 76, 87, 96, 123, 175, 193, 207, 225, 263, 302, 311, 316, 318, 366] [1, 36, 77, 88, 97, 124, 176, 194, 208, 226, 264, 303, 312, 317, 319, 367] [0, 2, 37, 78, 89, 98, 125, 177, 209, 227, 265, 304, 313, 318, 320, 368] [1, 3, 38, 79, 90, 99, 126, 178, 210, 228, 266, 305, 314, 319, 321, 369] [2, 4, 39, 80, 91, 100, 127, 179, 211, 229, 267, 306, 315, 320, 322, 370] [3, 5, 40, 81, 92, 101, 128, 180, 212, 230, 268, 307, 316, 321, 323, 371] [4, 6, 41, 82, 93, 102, 129, 181, 213, 231, 269, 308, 317, 322, 324, 372] [5, 7, 42, 83, 94, 103, 130, 182, 214, 232, 270, 309, 318, 323, 325, 373] [6, 8, 43, 84, 95, 104, 131, 183, 215, 233, 271, 310, 319, 324, 326, 374] [7, 9, 44, 85, 96, 105, 132, 184, 216, 234, 272, 311, 320, 325, 327, 375] [8, 10, 45, 86, 97, 106, 133, 185, 217, 235, 273, 312, 321, 326, 328, 376] [9, 11, 46, 87, 98, 107, 134, 186, 218, 236, 274, 313, 322, 327, 329, 377] [10, 12, 47, 88, 99, 108, 135, 187, 219, 237, 275, 314, 323, 328, 330, 378] [11, 13, 48, 89, 100, 109, 136, 188, 220, 238, 276, 315, 324, 329, 331, 379] [12, 14, 49, 90, 101, 110, 137, 189, 221, 239, 277, 316, 325, 330, 332, 380] [13, 15, 50, 91, 102, 111, 138, 190, 222, 240, 278, 317, 326, 331, 333, 381] [14, 16, 51, 92, 103, 112, 139, 191, 223, 241, 279, 318, 327, 332, 334, 382] [15, 17, 52, 93, 104, 113, 140, 192, 224, 242, 280, 319, 328, 333, 335, 383] [16, 18, 53, 94, 105, 114, 141, 193, 225, 243, 281, 320, 329, 334, 336, 384] [17, 19, 54, 95, 106, 115, 142, 194, 226, 244, 282, 321, 330, 335, 337, 385] [0, 18, 20, 55, 96, 107, 116, 143, 227, 245, 283, 322, 331, 336, 338, 386] [1, 19, 21, 56, 97, 108, 117, 144, 228, 246, 284, 323, 332, 337, 339, 387] [2, 20, 22, 57, 98, 109, 118, 145, 229, 247, 285, 324, 333, 338, 340, 388] [3, 21, 23, 58, 99, 110, 119, 146, 230, 248, 286, 325, 334, 339, 341, 389] [4, 22, 24, 59, 100, 111, 120, 147, 195, 231, 249, 287, 326, 335, 340, 342] [5, 23, 25, 60, 101, 112, 121, 148, 196, 232, 250, 288, 327, 336, 341, 343] [6, 24, 26, 61, 102, 113, 122, 149, 197, 233, 251, 289, 328, 337, 342, 344] [7, 25, 27, 62, 103, 114, 123, 150, 198, 234, 252, 290, 329, 338, 343, 345] [8, 26, 28, 63, 104, 115, 124, 151, 199, 235, 253, 291, 330, 339, 344, 346] [9, 27, 29, 64, 105, 116, 125, 152, 200, 236, 254, 292, 331, 340, 345, 347] [10, 28, 30, 65, 106, 117, 126, 153, 201, 237, 255, 293, 332, 341, 346, 348] [11, 29, 31, 66, 107, 118, 127, 154, 202, 238, 256, 294, 333, 342, 347, 349] [12, 30, 32, 67, 108, 119, 128, 155, 203, 239, 257, 295, 334, 343, 348, 350] [13, 31, 33, 68, 109, 120, 129, 156, 204, 240, 258, 296, 335, 344, 349, 351] [14, 32, 34, 69, 110, 121, 130, 157, 205, 241, 259, 297, 336, 345, 350, 352] [15, 33, 35, 70, 111, 122, 131, 158, 206, 242, 260, 298, 337, 346, 351, 353] [16, 34, 36, 71, 112, 123, 132, 159, 207, 243, 261, 299, 338, 347, 352, 354] [17, 35, 37, 72, 113, 124, 133, 160, 208, 244, 262, 300, 339, 348, 353, 355] [18, 36, 38, 73, 114, 125, 134, 161, 209, 245, 263, 301, 340, 349, 354, 356] [19, 37, 39, 74, 115, 126, 135, 162, 210, 246, 264, 302, 341, 350, 355, 357] [20, 38, 40, 75, 116, 127, 136, 163, 211, 247, 265, 303, 342, 351, 356, 358] [21, 39, 41, 76, 117, 128, 137, 164, 212, 248, 266, 304, 343, 352, 357, 359] [22, 40, 42, 77, 118, 129, 138, 165, 213, 249, 267, 305, 344, 353, 358, 360] [23, 41, 43, 78, 119, 130, 139, 166, 214, 250, 268, 306, 345, 354, 359, 361] [24, 42, 44, 79, 120, 131, 140, 167, 215, 251, 269, 307, 346, 355, 360, 362] [25, 43, 45, 80, 121, 132, 141, 168, 216, 252, 270, 308, 347, 356, 361, 363] [26, 44, 46, 81, 122, 133, 142, 169, 217, 253, 271, 309, 348, 357, 362, 364] [27, 45, 47, 82, 123, 134, 143, 170, 218, 254, 272, 310, 349, 358, 363, 365] [28, 46, 48, 83, 124, 135, 144, 171, 219, 255, 273, 311, 350, 359, 364, 366] [29, 47, 49, 84, 125, 136, 145, 172, 220, 256, 274, 312, 351, 360, 365, 367] [30, 48, 50, 85, 126, 137, 146, 173, 221, 257, 275, 313, 352, 361, 366, 368] [31, 49, 51, 86, 127, 138, 147, 174, 222, 258, 276, 314, 353, 362, 367, 369] [32, 50, 52, 87, 128, 139, 148, 175, 223, 259, 277, 315, 354, 363, 368, 370] [33, 51, 53, 88, 129, 140, 149, 176, 224, 260, 278, 316, 355, 364, 369, 371] [34, 52, 54, 89, 130, 141, 150, 177, 225, 261, 279, 317, 356, 365, 370, 372] [35, 53, 55, 90, 131, 142, 151, 178, 226, 262, 280, 318, 357, 366, 371, 373] [36, 54, 56, 91, 132, 143, 152, 179, 227, 263, 281, 319, 358, 367, 372, 374] [37, 55, 57, 92, 133, 144, 153, 180, 228, 264, 282, 320, 359, 368, 373, 375] [38, 56, 58, 93, 134, 145, 154, 181, 229, 265, 283, 321, 360, 369, 374, 376] [39, 57, 59, 94, 135, 146, 155, 182, 230, 266, 284, 322, 361, 370, 375, 377] [40, 58, 60, 95, 136, 147, 156, 183, 231, 267, 285, 323, 362, 371, 376, 378] [41, 59, 61, 96, 137, 148, 157, 184, 232, 268, 286, 324, 363, 372, 377, 379] [42, 60, 62, 97, 138, 149, 158, 185, 233, 269, 287, 325, 364, 373, 378, 380] [43, 61, 63, 98, 139, 150, 159, 186, 234, 270, 288, 326, 365, 374, 379, 381] [44, 62, 64, 99, 140, 151, 160, 187, 235, 271, 289, 327, 366, 375, 380, 382] [45, 63, 65, 100, 141, 152, 161, 188, 236, 272, 290, 328, 367, 376, 381, 383] [46, 64, 66, 101, 142, 153, 162, 189, 237, 273, 291, 329, 368, 377, 382, 384] [47, 65, 67, 102, 143, 154, 163, 190, 238, 274, 292, 330, 369, 378, 383, 385] [48, 66, 68, 103, 144, 155, 164, 191, 239, 275, 293, 331, 370, 379, 384, 386] [49, 67, 69, 104, 145, 156, 165, 192, 240, 276, 294, 332, 371, 380, 385, 387] [50, 68, 70, 105, 146, 157, 166, 193, 241, 277, 295, 333, 372, 381, 386, 388] [51, 69, 71, 106, 147, 158, 167, 194, 242, 278, 296, 334, 373, 382, 387, 389]
raw submission JSON