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[[472,122,14]] d ≤
n
472
k
122
d
14
kd²/n
50.661
w
8

Share this result

Distance

d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[3, 9, 13, 15, 121, 188, 195, 224, 287, 301, 305, 364, 372, 438]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[3, 24, 58, 101, 103, 121, 246, 287, 312, 346, 354, 393, 399, 438]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors Okada, Koki and Kasai, Kenta
provenance literature baseline
construction Pair-partition CPM-based CSS (arXiv:2607.14091v1): J=3 column weight, L=8 row weight, prime lift P=59, girth 6. H_X=(C(e_jl)), H_Z=(C(d_jl)) from pair-partition array M with paired-difference CSS orthogonality.
model classical construction (no AI model)
reference arXiv:2607.14091
date 2026
notes Literature baseline (arXiv:2607.14091v1), pair-partition CPM-based CSS. Distance witness from paper's exhaustive search, regenerated with the companion repo's own C++ verifier against the reconstructed exponents. Converted from kasaikenta/pair-partition-cpm-css-codes. Literature novelty confirmed by manual cross-check: arXiv:2607.14091v1 (Okada & Kasai, 15 Jul 2026) explicitly reports these [[n,k,d]] parameters in its abstract/Table 1 and establishes the distances by exhaustive low-weight exclusion with explicit non-stabilizer witnesses. The verifier's 'literature novelty UNVERIFIED' label is by-design (validate_candidate.py performs no literature lookup) and is intentionally not overridden.
family lifted product (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 177 · Z-checks 177
H_X (177 checks, sparse supports)
[28, 48, 86, 125, 129, 131, 186, 407] [36, 56, 94, 133, 137, 139, 194, 415] [44, 64, 102, 141, 145, 147, 202, 423] [52, 72, 110, 149, 153, 155, 210, 431] [60, 80, 118, 157, 161, 163, 218, 439] [68, 88, 126, 165, 169, 171, 226, 447] [76, 96, 134, 173, 177, 179, 234, 455] [84, 104, 142, 181, 185, 187, 242, 463] [92, 112, 150, 189, 193, 195, 250, 471] [7, 100, 120, 158, 197, 201, 203, 258] [15, 108, 128, 166, 205, 209, 211, 266] [23, 116, 136, 174, 213, 217, 219, 274] [31, 124, 144, 182, 221, 225, 227, 282] [39, 132, 152, 190, 229, 233, 235, 290] [47, 140, 160, 198, 237, 241, 243, 298] [55, 148, 168, 206, 245, 249, 251, 306] [63, 156, 176, 214, 253, 257, 259, 314] [71, 164, 184, 222, 261, 265, 267, 322] [79, 172, 192, 230, 269, 273, 275, 330] [87, 180, 200, 238, 277, 281, 283, 338] [95, 188, 208, 246, 285, 289, 291, 346] [103, 196, 216, 254, 293, 297, 299, 354] [111, 204, 224, 262, 301, 305, 307, 362] [119, 212, 232, 270, 309, 313, 315, 370] [127, 220, 240, 278, 317, 321, 323, 378] [135, 228, 248, 286, 325, 329, 331, 386] [143, 236, 256, 294, 333, 337, 339, 394] [151, 244, 264, 302, 341, 345, 347, 402] [159, 252, 272, 310, 349, 353, 355, 410] [167, 260, 280, 318, 357, 361, 363, 418] [175, 268, 288, 326, 365, 369, 371, 426] [183, 276, 296, 334, 373, 377, 379, 434] [191, 284, 304, 342, 381, 385, 387, 442] [199, 292, 312, 350, 389, 393, 395, 450] [207, 300, 320, 358, 397, 401, 403, 458] [215, 308, 328, 366, 405, 409, 411, 466] [2, 223, 316, 336, 374, 413, 417, 419] [10, 231, 324, 344, 382, 421, 425, 427] [18, 239, 332, 352, 390, 429, 433, 435] [26, 247, 340, 360, 398, 437, 441, 443] [34, 255, 348, 368, 406, 445, 449, 451] [42, 263, 356, 376, 414, 453, 457, 459] [50, 271, 364, 384, 422, 461, 465, 467] [1, 3, 58, 279, 372, 392, 430, 469] [5, 9, 11, 66, 287, 380, 400, 438] [13, 17, 19, 74, 295, 388, 408, 446] [21, 25, 27, 82, 303, 396, 416, 454] [29, 33, 35, 90, 311, 404, 424, 462] [37, 41, 43, 98, 319, 412, 432, 470] [6, 45, 49, 51, 106, 327, 420, 440] [14, 53, 57, 59, 114, 335, 428, 448] [22, 61, 65, 67, 122, 343, 436, 456] [30, 69, 73, 75, 130, 351, 444, 464] [0, 38, 77, 81, 83, 138, 359, 452] [8, 46, 85, 89, 91, 146, 367, 460] [16, 54, 93, 97, 99, 154, 375, 468] [4, 24, 62, 101, 105, 107, 162, 383] [12, 32, 70, 109, 113, 115, 170, 391] [20, 40, 78, 117, 121, 123, 178, 399] [24, 58, 103, 117, 246, 283, 409, 460] [32, 66, 111, 125, 254, 291, 417, 468] [4, 40, 74, 119, 133, 262, 299, 425] [12, 48, 82, 127, 141, 270, 307, 433] [20, 56, 90, 135, 149, 278, 315, 441] [28, 64, 98, 143, 157, 286, 323, 449] [36, 72, 106, 151, 165, 294, 331, 457] [44, 80, 114, 159, 173, 302, 339, 465] [1, 52, 88, 122, 167, 181, 310, 347] [9, 60, 96, 130, 175, 189, 318, 355] [17, 68, 104, 138, 183, 197, 326, 363] [25, 76, 112, 146, 191, 205, 334, 371] [33, 84, 120, 154, 199, 213, 342, 379] [41, 92, 128, 162, 207, 221, 350, 387] [49, 100, 136, 170, 215, 229, 358, 395] [57, 108, 144, 178, 223, 237, 366, 403] [65, 116, 152, 186, 231, 245, 374, 411] [73, 124, 160, 194, 239, 253, 382, 419] [81, 132, 168, 202, 247, 261, 390, 427] [89, 140, 176, 210, 255, 269, 398, 435] [97, 148, 184, 218, 263, 277, 406, 443] [105, 156, 192, 226, 271, 285, 414, 451] [113, 164, 200, 234, 279, 293, 422, 459] [121, 172, 208, 242, 287, 301, 430, 467] [3, 129, 180, 216, 250, 295, 309, 438] [11, 137, 188, 224, 258, 303, 317, 446] [19, 145, 196, 232, 266, 311, 325, 454] [27, 153, 204, 240, 274, 319, 333, 462] [35, 161, 212, 248, 282, 327, 341, 470] [6, 43, 169, 220, 256, 290, 335, 349] [14, 51, 177, 228, 264, 298, 343, 357] [22, 59, 185, 236, 272, 306, 351, 365] [30, 67, 193, 244, 280, 314, 359, 373] [38, 75, 201, 252, 288, 322, 367, 381] [46, 83, 209, 260, 296, 330, 375, 389] [54, 91, 217, 268, 304, 338, 383, 397] [62, 99, 225, 276, 312, 346, 391, 405] [70, 107, 233, 284, 320, 354, 399, 413] [78, 115, 241, 292, 328, 362, 407, 421] [86, 123, 249, 300, 336, 370, 415, 429] [94, 131, 257, 308, 344, 378, 423, 437] [102, 139, 265, 316, 352, 386, 431, 445] [110, 147, 273, 324, 360, 394, 439, 453] [118, 155, 281, 332, 368, 402, 447, 461] [126, 163, 289, 340, 376, 410, 455, 469] [5, 134, 171, 297, 348, 384, 418, 463] [13, 142, 179, 305, 356, 392, 426, 471] [7, 21, 150, 187, 313, 364, 400, 434] [15, 29, 158, 195, 321, 372, 408, 442] [23, 37, 166, 203, 329, 380, 416, 450] [31, 45, 174, 211, 337, 388, 424, 458] [39, 53, 182, 219, 345, 396, 432, 466] [2, 47, 61, 190, 227, 353, 404, 440] [10, 55, 69, 198, 235, 361, 412, 448] [18, 63, 77, 206, 243, 369, 420, 456] [26, 71, 85, 214, 251, 377, 428, 464] [0, 34, 79, 93, 222, 259, 385, 436] [8, 42, 87, 101, 230, 267, 393, 444] [16, 50, 95, 109, 238, 275, 401, 452] [122, 165, 214, 223, 292, 299, 361, 432] [130, 173, 222, 231, 300, 307, 369, 440] [138, 181, 230, 239, 308, 315, 377, 448] [146, 189, 238, 247, 316, 323, 385, 456] [154, 197, 246, 255, 324, 331, 393, 464] [0, 162, 205, 254, 263, 332, 339, 401] [8, 170, 213, 262, 271, 340, 347, 409] [16, 178, 221, 270, 279, 348, 355, 417] [24, 186, 229, 278, 287, 356, 363, 425] [32, 194, 237, 286, 295, 364, 371, 433] [40, 202, 245, 294, 303, 372, 379, 441] [48, 210, 253, 302, 311, 380, 387, 449] [56, 218, 261, 310, 319, 388, 395, 457] [64, 226, 269, 318, 327, 396, 403, 465] [1, 72, 234, 277, 326, 335, 404, 411] [9, 80, 242, 285, 334, 343, 412, 419] [17, 88, 250, 293, 342, 351, 420, 427] [25, 96, 258, 301, 350, 359, 428, 435] [33, 104, 266, 309, 358, 367, 436, 443] [41, 112, 274, 317, 366, 375, 444, 451] [49, 120, 282, 325, 374, 383, 452, 459] [57, 128, 290, 333, 382, 391, 460, 467] [3, 65, 136, 298, 341, 390, 399, 468] [4, 11, 73, 144, 306, 349, 398, 407] [12, 19, 81, 152, 314, 357, 406, 415] [20, 27, 89, 160, 322, 365, 414, 423] [28, 35, 97, 168, 330, 373, 422, 431] [36, 43, 105, 176, 338, 381, 430, 439] [44, 51, 113, 184, 346, 389, 438, 447] [52, 59, 121, 192, 354, 397, 446, 455] [60, 67, 129, 200, 362, 405, 454, 463] [68, 75, 137, 208, 370, 413, 462, 471] [7, 76, 83, 145, 216, 378, 421, 470] [6, 15, 84, 91, 153, 224, 386, 429] [14, 23, 92, 99, 161, 232, 394, 437] [22, 31, 100, 107, 169, 240, 402, 445] [30, 39, 108, 115, 177, 248, 410, 453] [38, 47, 116, 123, 185, 256, 418, 461] [46, 55, 124, 131, 193, 264, 426, 469] [5, 54, 63, 132, 139, 201, 272, 434] [13, 62, 71, 140, 147, 209, 280, 442] [21, 70, 79, 148, 155, 217, 288, 450] [29, 78, 87, 156, 163, 225, 296, 458] [37, 86, 95, 164, 171, 233, 304, 466] [2, 45, 94, 103, 172, 179, 241, 312] [10, 53, 102, 111, 180, 187, 249, 320] [18, 61, 110, 119, 188, 195, 257, 328] [26, 69, 118, 127, 196, 203, 265, 336] [34, 77, 126, 135, 204, 211, 273, 344] [42, 85, 134, 143, 212, 219, 281, 352] [50, 93, 142, 151, 220, 227, 289, 360] [58, 101, 150, 159, 228, 235, 297, 368] [66, 109, 158, 167, 236, 243, 305, 376] [74, 117, 166, 175, 244, 251, 313, 384] [82, 125, 174, 183, 252, 259, 321, 392] [90, 133, 182, 191, 260, 267, 329, 400] [98, 141, 190, 199, 268, 275, 337, 408] [106, 149, 198, 207, 276, 283, 345, 416] [114, 157, 206, 215, 284, 291, 353, 424]
H_Z (177 checks, sparse supports)
[22, 67, 185, 244, 384, 402, 461, 463] [30, 75, 193, 252, 392, 410, 469, 471] [5, 7, 38, 83, 201, 260, 400, 418] [13, 15, 46, 91, 209, 268, 408, 426] [21, 23, 54, 99, 217, 276, 416, 434] [29, 31, 62, 107, 225, 284, 424, 442] [37, 39, 70, 115, 233, 292, 432, 450] [45, 47, 78, 123, 241, 300, 440, 458] [53, 55, 86, 131, 249, 308, 448, 466] [2, 61, 63, 94, 139, 257, 316, 456] [10, 69, 71, 102, 147, 265, 324, 464] [0, 18, 77, 79, 110, 155, 273, 332] [8, 26, 85, 87, 118, 163, 281, 340] [16, 34, 93, 95, 126, 171, 289, 348] [24, 42, 101, 103, 134, 179, 297, 356] [32, 50, 109, 111, 142, 187, 305, 364] [40, 58, 117, 119, 150, 195, 313, 372] [48, 66, 125, 127, 158, 203, 321, 380] [56, 74, 133, 135, 166, 211, 329, 388] [64, 82, 141, 143, 174, 219, 337, 396] [72, 90, 149, 151, 182, 227, 345, 404] [80, 98, 157, 159, 190, 235, 353, 412] [88, 106, 165, 167, 198, 243, 361, 420] [96, 114, 173, 175, 206, 251, 369, 428] [104, 122, 181, 183, 214, 259, 377, 436] [112, 130, 189, 191, 222, 267, 385, 444] [120, 138, 197, 199, 230, 275, 393, 452] [128, 146, 205, 207, 238, 283, 401, 460] [136, 154, 213, 215, 246, 291, 409, 468] [4, 144, 162, 221, 223, 254, 299, 417] [12, 152, 170, 229, 231, 262, 307, 425] [20, 160, 178, 237, 239, 270, 315, 433] [28, 168, 186, 245, 247, 278, 323, 441] [36, 176, 194, 253, 255, 286, 331, 449] [44, 184, 202, 261, 263, 294, 339, 457] [52, 192, 210, 269, 271, 302, 347, 465] [1, 60, 200, 218, 277, 279, 310, 355] [9, 68, 208, 226, 285, 287, 318, 363] [17, 76, 216, 234, 293, 295, 326, 371] [25, 84, 224, 242, 301, 303, 334, 379] [33, 92, 232, 250, 309, 311, 342, 387] [41, 100, 240, 258, 317, 319, 350, 395] [49, 108, 248, 266, 325, 327, 358, 403] [57, 116, 256, 274, 333, 335, 366, 411] [65, 124, 264, 282, 341, 343, 374, 419] [73, 132, 272, 290, 349, 351, 382, 427] [81, 140, 280, 298, 357, 359, 390, 435] [89, 148, 288, 306, 365, 367, 398, 443] [97, 156, 296, 314, 373, 375, 406, 451] [105, 164, 304, 322, 381, 383, 414, 459] [113, 172, 312, 330, 389, 391, 422, 467] [3, 121, 180, 320, 338, 397, 399, 430] [11, 129, 188, 328, 346, 405, 407, 438] [19, 137, 196, 336, 354, 413, 415, 446] [27, 145, 204, 344, 362, 421, 423, 454] [35, 153, 212, 352, 370, 429, 431, 462] [43, 161, 220, 360, 378, 437, 439, 470] [6, 51, 169, 228, 368, 386, 445, 447] [14, 59, 177, 236, 376, 394, 453, 455] [21, 25, 191, 350, 380, 387, 400, 450] [29, 33, 199, 358, 388, 395, 408, 458] [37, 41, 207, 366, 396, 403, 416, 466] [2, 45, 49, 215, 374, 404, 411, 424] [10, 53, 57, 223, 382, 412, 419, 432] [18, 61, 65, 231, 390, 420, 427, 440] [26, 69, 73, 239, 398, 428, 435, 448] [34, 77, 81, 247, 406, 436, 443, 456] [42, 85, 89, 255, 414, 444, 451, 464] [0, 50, 93, 97, 263, 422, 452, 459] [8, 58, 101, 105, 271, 430, 460, 467] [3, 16, 66, 109, 113, 279, 438, 468] [4, 11, 24, 74, 117, 121, 287, 446] [12, 19, 32, 82, 125, 129, 295, 454] [20, 27, 40, 90, 133, 137, 303, 462] [28, 35, 48, 98, 141, 145, 311, 470] [6, 36, 43, 56, 106, 149, 153, 319] [14, 44, 51, 64, 114, 157, 161, 327] [22, 52, 59, 72, 122, 165, 169, 335] [30, 60, 67, 80, 130, 173, 177, 343] [38, 68, 75, 88, 138, 181, 185, 351] [46, 76, 83, 96, 146, 189, 193, 359] [54, 84, 91, 104, 154, 197, 201, 367] [62, 92, 99, 112, 162, 205, 209, 375] [70, 100, 107, 120, 170, 213, 217, 383] [78, 108, 115, 128, 178, 221, 225, 391] [86, 116, 123, 136, 186, 229, 233, 399] [94, 124, 131, 144, 194, 237, 241, 407] [102, 132, 139, 152, 202, 245, 249, 415] [110, 140, 147, 160, 210, 253, 257, 423] [118, 148, 155, 168, 218, 261, 265, 431] [126, 156, 163, 176, 226, 269, 273, 439] [134, 164, 171, 184, 234, 277, 281, 447] [142, 172, 179, 192, 242, 285, 289, 455] [150, 180, 187, 200, 250, 293, 297, 463] [158, 188, 195, 208, 258, 301, 305, 471] [7, 166, 196, 203, 216, 266, 309, 313] [15, 174, 204, 211, 224, 274, 317, 321] [23, 182, 212, 219, 232, 282, 325, 329] [31, 190, 220, 227, 240, 290, 333, 337] [39, 198, 228, 235, 248, 298, 341, 345] [47, 206, 236, 243, 256, 306, 349, 353] [55, 214, 244, 251, 264, 314, 357, 361] [63, 222, 252, 259, 272, 322, 365, 369] [71, 230, 260, 267, 280, 330, 373, 377] [79, 238, 268, 275, 288, 338, 381, 385] [87, 246, 276, 283, 296, 346, 389, 393] [95, 254, 284, 291, 304, 354, 397, 401] [103, 262, 292, 299, 312, 362, 405, 409] [111, 270, 300, 307, 320, 370, 413, 417] [119, 278, 308, 315, 328, 378, 421, 425] [127, 286, 316, 323, 336, 386, 429, 433] [135, 294, 324, 331, 344, 394, 437, 441] [143, 302, 332, 339, 352, 402, 445, 449] [151, 310, 340, 347, 360, 410, 453, 457] [159, 318, 348, 355, 368, 418, 461, 465] [1, 167, 326, 356, 363, 376, 426, 469] [5, 9, 175, 334, 364, 371, 384, 434] [13, 17, 183, 342, 372, 379, 392, 442] [28, 46, 64, 89, 131, 269, 330, 423] [36, 54, 72, 97, 139, 277, 338, 431] [44, 62, 80, 105, 147, 285, 346, 439] [52, 70, 88, 113, 155, 293, 354, 447] [60, 78, 96, 121, 163, 301, 362, 455] [68, 86, 104, 129, 171, 309, 370, 463] [76, 94, 112, 137, 179, 317, 378, 471] [7, 84, 102, 120, 145, 187, 325, 386] [15, 92, 110, 128, 153, 195, 333, 394] [23, 100, 118, 136, 161, 203, 341, 402] [31, 108, 126, 144, 169, 211, 349, 410] [39, 116, 134, 152, 177, 219, 357, 418] [47, 124, 142, 160, 185, 227, 365, 426] [55, 132, 150, 168, 193, 235, 373, 434] [63, 140, 158, 176, 201, 243, 381, 442] [71, 148, 166, 184, 209, 251, 389, 450] [79, 156, 174, 192, 217, 259, 397, 458] [87, 164, 182, 200, 225, 267, 405, 466] [2, 95, 172, 190, 208, 233, 275, 413] [10, 103, 180, 198, 216, 241, 283, 421] [18, 111, 188, 206, 224, 249, 291, 429] [26, 119, 196, 214, 232, 257, 299, 437] [34, 127, 204, 222, 240, 265, 307, 445] [42, 135, 212, 230, 248, 273, 315, 453] [50, 143, 220, 238, 256, 281, 323, 461] [58, 151, 228, 246, 264, 289, 331, 469] [5, 66, 159, 236, 254, 272, 297, 339] [13, 74, 167, 244, 262, 280, 305, 347] [21, 82, 175, 252, 270, 288, 313, 355] [29, 90, 183, 260, 278, 296, 321, 363] [37, 98, 191, 268, 286, 304, 329, 371] [45, 106, 199, 276, 294, 312, 337, 379] [53, 114, 207, 284, 302, 320, 345, 387] [61, 122, 215, 292, 310, 328, 353, 395] [69, 130, 223, 300, 318, 336, 361, 403] [77, 138, 231, 308, 326, 344, 369, 411] [85, 146, 239, 316, 334, 352, 377, 419] [93, 154, 247, 324, 342, 360, 385, 427] [101, 162, 255, 332, 350, 368, 393, 435] [109, 170, 263, 340, 358, 376, 401, 443] [117, 178, 271, 348, 366, 384, 409, 451] [125, 186, 279, 356, 374, 392, 417, 459] [133, 194, 287, 364, 382, 400, 425, 467] [3, 141, 202, 295, 372, 390, 408, 433] [11, 149, 210, 303, 380, 398, 416, 441] [19, 157, 218, 311, 388, 406, 424, 449] [27, 165, 226, 319, 396, 414, 432, 457] [35, 173, 234, 327, 404, 422, 440, 465] [1, 43, 181, 242, 335, 412, 430, 448] [9, 51, 189, 250, 343, 420, 438, 456] [17, 59, 197, 258, 351, 428, 446, 464] [0, 25, 67, 205, 266, 359, 436, 454] [8, 33, 75, 213, 274, 367, 444, 462] [16, 41, 83, 221, 282, 375, 452, 470] [6, 24, 49, 91, 229, 290, 383, 460] [14, 32, 57, 99, 237, 298, 391, 468] [4, 22, 40, 65, 107, 245, 306, 399] [12, 30, 48, 73, 115, 253, 314, 407] [20, 38, 56, 81, 123, 261, 322, 415]
raw submission JSON