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[[392,102,14]] d ≤
n
392
k
102
d
14
kd²/n
51.0
w
8
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 14 · witness weight 14 (claimed exact)
witness operator (support, 14 qubits)
[0, 31, 70, 96, 98, 99, 129, 197, 236, 264, 272, 273, 337, 369]
d_Z 14 · witness weight 14 (claimed exact)
witness operator (support, 14 qubits)
[0, 47, 53, 66, 75, 93, 110, 132, 176, 185, 220, 229, 315, 340]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 8 · H_Z 8
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×392 (2,4)×4116 (3,3)×49 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 392 (2,4): 4116 (3,3): 49 (3,5): 57477 (3,7): 8232
trapping sets H_Z (1,3)×392 (2,4)×4116 (3,3)×49 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 392 (2,4): 4116 (3,3): 49 (3,5): 57477 (3,7): 8232

Construction & provenance

authors Okada, Koki and Kasai, Kenta
provenance literature baseline
construction Pair-partition GPM (generalized permutation matrix) CSS construction over C7 x C7 with regular action (arXiv:2607.14091), instance gpm_pp_c7c7_frontier: group order 49, (J,L)=(3,8), n=8*49=392, 147 checks per side, column weight 3, row weight 8, no Tanner 4-cycles. H_X and H_Z are taken verbatim from the authors' NumPy archive https://kasai.ict.eng.isct.ac.jp/pair_partition_cpm_css_codes_data/distance_records/gpm_pp_frontier_20260811/c7c7_d14/gpm_pp_c7c7_frontier.npz (sha256 ee438d409916e85abc0a386d0bdf9f5fb4abf092703996df851005444995d23b), which also carries the pair-partition array M and GPM exponent arrays E, D used to build them.
model classical construction (no AI model)
reference arXiv:2607.14091
date 2026
notes Literature baseline, seeded 2026-08-14. Listed as the GPM-PP instance [[392,102,14]] (C7 x C7) in the primary table of the Okada-Kasai pair-partition CPM catalogue (arXiv:2607.14091; catalogue page updated 2026-08-13). The authors publish an exact-distance certificate (https://kasai.ict.eng.isct.ac.jp/pair_partition_cpm_css_codes_data/distance_records/gpm_pp_frontier_20260811/c7c7_d14/final_certificate.json, sha256 7f4fb595507f6383704aa6cf90b0996bbf70765fc7e3d9640ab8cac2e11bf109): a complete connected-support search through weight 12 on both sides plus explicit weight-14 logicals, which are the witnesses here, re-verified in this repo (kernel membership, row-space non-membership). Marked exact on the strength of that public certificate; per board policy the entry still displays as an upper bound until this repo's own certification. Reproduction was validated end to end: the archive's H_X/H_Z are CSS-orthogonal, (3,8)-regular, GF(2) ranks 145/145 (k=102), matching the certificate's matrix audit.
family pair-partition CPM (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 147 (max weight 8) · Z-checks 147 (max weight 8)
H_X (147 checks, sparse supports)
[5, 73, 105, 148, 241, 268, 342, 390] [6, 74, 106, 149, 242, 269, 336, 391] [0, 75, 107, 150, 243, 270, 337, 385] [1, 76, 108, 151, 244, 271, 338, 386] [2, 70, 109, 152, 238, 272, 339, 387] [3, 71, 110, 153, 239, 266, 340, 388] [4, 72, 111, 147, 240, 267, 341, 389] [12, 80, 112, 155, 199, 275, 300, 348] [13, 81, 113, 156, 200, 276, 294, 349] [7, 82, 114, 157, 201, 277, 295, 343] [8, 83, 115, 158, 202, 278, 296, 344] [9, 77, 116, 159, 196, 279, 297, 345] [10, 78, 117, 160, 197, 273, 298, 346] [11, 79, 118, 154, 198, 274, 299, 347] [19, 87, 119, 162, 206, 282, 307, 355] [20, 88, 120, 163, 207, 283, 301, 356] [14, 89, 121, 164, 208, 284, 302, 350] [15, 90, 122, 165, 209, 285, 303, 351] [16, 84, 123, 166, 203, 286, 304, 352] [17, 85, 124, 167, 204, 280, 305, 353] [18, 86, 125, 161, 205, 281, 306, 354] [26, 94, 126, 169, 213, 289, 314, 362] [27, 95, 127, 170, 214, 290, 308, 363] [21, 96, 128, 171, 215, 291, 309, 357] [22, 97, 129, 172, 216, 292, 310, 358] [23, 91, 130, 173, 210, 293, 311, 359] [24, 92, 131, 174, 211, 287, 312, 360] [25, 93, 132, 168, 212, 288, 313, 361] [33, 52, 133, 176, 220, 247, 321, 369] [34, 53, 134, 177, 221, 248, 315, 370] [28, 54, 135, 178, 222, 249, 316, 364] [29, 55, 136, 179, 223, 250, 317, 365] [30, 49, 137, 180, 217, 251, 318, 366] [31, 50, 138, 181, 218, 245, 319, 367] [32, 51, 139, 175, 219, 246, 320, 368] [40, 59, 140, 183, 227, 254, 328, 376] [41, 60, 141, 184, 228, 255, 322, 377] [35, 61, 142, 185, 229, 256, 323, 371] [36, 62, 143, 186, 230, 257, 324, 372] [37, 56, 144, 187, 224, 258, 325, 373] [38, 57, 145, 188, 225, 252, 326, 374] [39, 58, 146, 182, 226, 253, 327, 375] [47, 66, 98, 190, 234, 261, 335, 383] [48, 67, 99, 191, 235, 262, 329, 384] [42, 68, 100, 192, 236, 263, 330, 378] [43, 69, 101, 193, 237, 264, 331, 379] [44, 63, 102, 194, 231, 265, 332, 380] [45, 64, 103, 195, 232, 259, 333, 381] [46, 65, 104, 189, 233, 260, 334, 382] [5, 75, 132, 183, 204, 290, 324, 350] [6, 76, 126, 184, 205, 291, 325, 351] [0, 70, 127, 185, 206, 292, 326, 352] [1, 71, 128, 186, 207, 293, 327, 353] [2, 72, 129, 187, 208, 287, 328, 354] [3, 73, 130, 188, 209, 288, 322, 355] [4, 74, 131, 182, 203, 289, 323, 356] [12, 82, 139, 190, 211, 248, 331, 357] [13, 83, 133, 191, 212, 249, 332, 358] [7, 77, 134, 192, 213, 250, 333, 359] [8, 78, 135, 193, 214, 251, 334, 360] [9, 79, 136, 194, 215, 245, 335, 361] [10, 80, 137, 195, 216, 246, 329, 362] [11, 81, 138, 189, 210, 247, 330, 363] [19, 89, 146, 148, 218, 255, 338, 364] [20, 90, 140, 149, 219, 256, 339, 365] [14, 84, 141, 150, 220, 257, 340, 366] [15, 85, 142, 151, 221, 258, 341, 367] [16, 86, 143, 152, 222, 252, 342, 368] [17, 87, 144, 153, 223, 253, 336, 369] [18, 88, 145, 147, 217, 254, 337, 370] [26, 96, 104, 155, 225, 262, 296, 371] [27, 97, 98, 156, 226, 263, 297, 372] [21, 91, 99, 157, 227, 264, 298, 373] [22, 92, 100, 158, 228, 265, 299, 374] [23, 93, 101, 159, 229, 259, 300, 375] [24, 94, 102, 160, 230, 260, 294, 376] [25, 95, 103, 154, 224, 261, 295, 377] [33, 54, 111, 162, 232, 269, 303, 378] [34, 55, 105, 163, 233, 270, 304, 379] [28, 49, 106, 164, 234, 271, 305, 380] [29, 50, 107, 165, 235, 272, 306, 381] [30, 51, 108, 166, 236, 266, 307, 382] [31, 52, 109, 167, 237, 267, 301, 383] [32, 53, 110, 161, 231, 268, 302, 384] [40, 61, 118, 169, 239, 276, 310, 385] [41, 62, 112, 170, 240, 277, 311, 386] [35, 56, 113, 171, 241, 278, 312, 387] [36, 57, 114, 172, 242, 279, 313, 388] [37, 58, 115, 173, 243, 273, 314, 389] [38, 59, 116, 174, 244, 274, 308, 390] [39, 60, 117, 168, 238, 275, 309, 391] [47, 68, 125, 176, 197, 283, 317, 343] [48, 69, 119, 177, 198, 284, 318, 344] [42, 63, 120, 178, 199, 285, 319, 345] [43, 64, 121, 179, 200, 286, 320, 346] [44, 65, 122, 180, 201, 280, 321, 347] [45, 66, 123, 181, 202, 281, 315, 348] [46, 67, 124, 175, 196, 282, 316, 349] [8, 55, 118, 187, 209, 246, 301, 375] [9, 49, 112, 188, 203, 247, 302, 376] [10, 50, 113, 182, 204, 248, 303, 377] [11, 51, 114, 183, 205, 249, 304, 371] [12, 52, 115, 184, 206, 250, 305, 372] [13, 53, 116, 185, 207, 251, 306, 373] [7, 54, 117, 186, 208, 245, 307, 374] [15, 62, 125, 194, 216, 253, 308, 382] [16, 56, 119, 195, 210, 254, 309, 383] [17, 57, 120, 189, 211, 255, 310, 384] [18, 58, 121, 190, 212, 256, 311, 378] [19, 59, 122, 191, 213, 257, 312, 379] [20, 60, 123, 192, 214, 258, 313, 380] [14, 61, 124, 193, 215, 252, 314, 381] [22, 69, 132, 152, 223, 260, 315, 389] [23, 63, 126, 153, 217, 261, 316, 390] [24, 64, 127, 147, 218, 262, 317, 391] [25, 65, 128, 148, 219, 263, 318, 385] [26, 66, 129, 149, 220, 264, 319, 386] [27, 67, 130, 150, 221, 265, 320, 387] [21, 68, 131, 151, 222, 259, 321, 388] [29, 76, 139, 159, 230, 267, 322, 347] [30, 70, 133, 160, 224, 268, 323, 348] [31, 71, 134, 154, 225, 269, 324, 349] [32, 72, 135, 155, 226, 270, 325, 343] [33, 73, 136, 156, 227, 271, 326, 344] [34, 74, 137, 157, 228, 272, 327, 345] [28, 75, 138, 158, 229, 266, 328, 346] [36, 83, 146, 166, 237, 274, 329, 354] [37, 77, 140, 167, 231, 275, 330, 355] [38, 78, 141, 161, 232, 276, 331, 356] [39, 79, 142, 162, 233, 277, 332, 350] [40, 80, 143, 163, 234, 278, 333, 351] [41, 81, 144, 164, 235, 279, 334, 352] [35, 82, 145, 165, 236, 273, 335, 353] [43, 90, 104, 173, 244, 281, 336, 361] [44, 84, 98, 174, 238, 282, 337, 362] [45, 85, 99, 168, 239, 283, 338, 363] [46, 86, 100, 169, 240, 284, 339, 357] [47, 87, 101, 170, 241, 285, 340, 358] [48, 88, 102, 171, 242, 286, 341, 359] [42, 89, 103, 172, 243, 280, 342, 360] [1, 97, 111, 180, 202, 288, 294, 368] [2, 91, 105, 181, 196, 289, 295, 369] [3, 92, 106, 175, 197, 290, 296, 370] [4, 93, 107, 176, 198, 291, 297, 364] [5, 94, 108, 177, 199, 292, 298, 365] [6, 95, 109, 178, 200, 293, 299, 366] [0, 96, 110, 179, 201, 287, 300, 367]
H_Z (147 checks, sparse supports)
[3, 92, 110, 188, 208, 287, 302, 374] [4, 93, 111, 182, 209, 288, 303, 375] [5, 94, 105, 183, 203, 289, 304, 376] [6, 95, 106, 184, 204, 290, 305, 377] [0, 96, 107, 185, 205, 291, 306, 371] [1, 97, 108, 186, 206, 292, 307, 372] [2, 91, 109, 187, 207, 293, 301, 373] [10, 50, 117, 195, 215, 245, 309, 381] [11, 51, 118, 189, 216, 246, 310, 382] [12, 52, 112, 190, 210, 247, 311, 383] [13, 53, 113, 191, 211, 248, 312, 384] [7, 54, 114, 192, 212, 249, 313, 378] [8, 55, 115, 193, 213, 250, 314, 379] [9, 49, 116, 194, 214, 251, 308, 380] [17, 57, 124, 153, 222, 252, 316, 388] [18, 58, 125, 147, 223, 253, 317, 389] [19, 59, 119, 148, 217, 254, 318, 390] [20, 60, 120, 149, 218, 255, 319, 391] [14, 61, 121, 150, 219, 256, 320, 385] [15, 62, 122, 151, 220, 257, 321, 386] [16, 56, 123, 152, 221, 258, 315, 387] [24, 64, 131, 160, 229, 259, 323, 346] [25, 65, 132, 154, 230, 260, 324, 347] [26, 66, 126, 155, 224, 261, 325, 348] [27, 67, 127, 156, 225, 262, 326, 349] [21, 68, 128, 157, 226, 263, 327, 343] [22, 69, 129, 158, 227, 264, 328, 344] [23, 63, 130, 159, 228, 265, 322, 345] [31, 71, 138, 167, 236, 266, 330, 353] [32, 72, 139, 161, 237, 267, 331, 354] [33, 73, 133, 162, 231, 268, 332, 355] [34, 74, 134, 163, 232, 269, 333, 356] [28, 75, 135, 164, 233, 270, 334, 350] [29, 76, 136, 165, 234, 271, 335, 351] [30, 70, 137, 166, 235, 272, 329, 352] [38, 78, 145, 174, 243, 273, 337, 360] [39, 79, 146, 168, 244, 274, 338, 361] [40, 80, 140, 169, 238, 275, 339, 362] [41, 81, 141, 170, 239, 276, 340, 363] [35, 82, 142, 171, 240, 277, 341, 357] [36, 83, 143, 172, 241, 278, 342, 358] [37, 77, 144, 173, 242, 279, 336, 359] [45, 85, 103, 181, 201, 280, 295, 367] [46, 86, 104, 175, 202, 281, 296, 368] [47, 87, 98, 176, 196, 282, 297, 369] [48, 88, 99, 177, 197, 283, 298, 370] [42, 89, 100, 178, 198, 284, 299, 364] [43, 90, 101, 179, 199, 285, 300, 365] [44, 84, 102, 180, 200, 286, 294, 366] [23, 91, 101, 153, 220, 264, 340, 369] [24, 92, 102, 147, 221, 265, 341, 370] [25, 93, 103, 148, 222, 259, 342, 364] [26, 94, 104, 149, 223, 260, 336, 365] [27, 95, 98, 150, 217, 261, 337, 366] [21, 96, 99, 151, 218, 262, 338, 367] [22, 97, 100, 152, 219, 263, 339, 368] [30, 49, 108, 160, 227, 271, 298, 376] [31, 50, 109, 154, 228, 272, 299, 377] [32, 51, 110, 155, 229, 266, 300, 371] [33, 52, 111, 156, 230, 267, 294, 372] [34, 53, 105, 157, 224, 268, 295, 373] [28, 54, 106, 158, 225, 269, 296, 374] [29, 55, 107, 159, 226, 270, 297, 375] [37, 56, 115, 167, 234, 278, 305, 383] [38, 57, 116, 161, 235, 279, 306, 384] [39, 58, 117, 162, 236, 273, 307, 378] [40, 59, 118, 163, 237, 274, 301, 379] [41, 60, 112, 164, 231, 275, 302, 380] [35, 61, 113, 165, 232, 276, 303, 381] [36, 62, 114, 166, 233, 277, 304, 382] [44, 63, 122, 174, 241, 285, 312, 390] [45, 64, 123, 168, 242, 286, 313, 391] [46, 65, 124, 169, 243, 280, 314, 385] [47, 66, 125, 170, 244, 281, 308, 386] [48, 67, 119, 171, 238, 282, 309, 387] [42, 68, 120, 172, 239, 283, 310, 388] [43, 69, 121, 173, 240, 284, 311, 389] [2, 70, 129, 181, 199, 292, 319, 348] [3, 71, 130, 175, 200, 293, 320, 349] [4, 72, 131, 176, 201, 287, 321, 343] [5, 73, 132, 177, 202, 288, 315, 344] [6, 74, 126, 178, 196, 289, 316, 345] [0, 75, 127, 179, 197, 290, 317, 346] [1, 76, 128, 180, 198, 291, 318, 347] [9, 77, 136, 188, 206, 250, 326, 355] [10, 78, 137, 182, 207, 251, 327, 356] [11, 79, 138, 183, 208, 245, 328, 350] [12, 80, 139, 184, 209, 246, 322, 351] [13, 81, 133, 185, 203, 247, 323, 352] [7, 82, 134, 186, 204, 248, 324, 353] [8, 83, 135, 187, 205, 249, 325, 354] [16, 84, 143, 195, 213, 257, 333, 362] [17, 85, 144, 189, 214, 258, 334, 363] [18, 86, 145, 190, 215, 252, 335, 357] [19, 87, 146, 191, 216, 253, 329, 358] [20, 88, 140, 192, 210, 254, 330, 359] [14, 89, 141, 193, 211, 255, 331, 360] [15, 90, 142, 194, 212, 256, 332, 361] [27, 97, 130, 170, 212, 288, 311, 358] [21, 91, 131, 171, 213, 289, 312, 359] [22, 92, 132, 172, 214, 290, 313, 360] [23, 93, 126, 173, 215, 291, 314, 361] [24, 94, 127, 174, 216, 292, 308, 362] [25, 95, 128, 168, 210, 293, 309, 363] [26, 96, 129, 169, 211, 287, 310, 357] [34, 55, 137, 177, 219, 246, 318, 365] [28, 49, 138, 178, 220, 247, 319, 366] [29, 50, 139, 179, 221, 248, 320, 367] [30, 51, 133, 180, 222, 249, 321, 368] [31, 52, 134, 181, 223, 250, 315, 369] [32, 53, 135, 175, 217, 251, 316, 370] [33, 54, 136, 176, 218, 245, 317, 364] [41, 62, 144, 184, 226, 253, 325, 372] [35, 56, 145, 185, 227, 254, 326, 373] [36, 57, 146, 186, 228, 255, 327, 374] [37, 58, 140, 187, 229, 256, 328, 375] [38, 59, 141, 188, 230, 257, 322, 376] [39, 60, 142, 182, 224, 258, 323, 377] [40, 61, 143, 183, 225, 252, 324, 371] [48, 69, 102, 191, 233, 260, 332, 379] [42, 63, 103, 192, 234, 261, 333, 380] [43, 64, 104, 193, 235, 262, 334, 381] [44, 65, 98, 194, 236, 263, 335, 382] [45, 66, 99, 195, 237, 264, 329, 383] [46, 67, 100, 189, 231, 265, 330, 384] [47, 68, 101, 190, 232, 259, 331, 378] [6, 76, 109, 149, 240, 267, 339, 386] [0, 70, 110, 150, 241, 268, 340, 387] [1, 71, 111, 151, 242, 269, 341, 388] [2, 72, 105, 152, 243, 270, 342, 389] [3, 73, 106, 153, 244, 271, 336, 390] [4, 74, 107, 147, 238, 272, 337, 391] [5, 75, 108, 148, 239, 266, 338, 385] [13, 83, 116, 156, 198, 274, 297, 344] [7, 77, 117, 157, 199, 275, 298, 345] [8, 78, 118, 158, 200, 276, 299, 346] [9, 79, 112, 159, 201, 277, 300, 347] [10, 80, 113, 160, 202, 278, 294, 348] [11, 81, 114, 154, 196, 279, 295, 349] [12, 82, 115, 155, 197, 273, 296, 343] [20, 90, 123, 163, 205, 281, 304, 351] [14, 84, 124, 164, 206, 282, 305, 352] [15, 85, 125, 165, 207, 283, 306, 353] [16, 86, 119, 166, 208, 284, 307, 354] [17, 87, 120, 167, 209, 285, 301, 355] [18, 88, 121, 161, 203, 286, 302, 356] [19, 89, 122, 162, 204, 280, 303, 350]
Code ID 392-102-14 · download JSON · raw on GitHub