Best Known (107, ∞, s)-Nets in Base 7
(107, ∞, 88)-Net over F7 — Constructive and digital
Digital (107, m, 88)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (107, 87)-sequence over F7, using
- Niederreiter–Xing sequence construction III based on function field F2/F7 with g(F2) = 21, N(F2) ≥ 2, and N(F2) + N2(F2) ≥ 88 from GarcÃa–Stichtenoth tower as constant field extension [i]
(107, ∞, 105)-Net over F7 — Digital
Digital (107, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (107, 104)-sequence over F7, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 43 and N(F) ≥ 105, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
(107, ∞, 666)-Net in Base 7 — Upper bound on s
There is no (107, m, 667)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (107, 1997, 667)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71997, 667, S7, 3, 1890), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 92636 426004 826295 821073 567763 094703 375235 549504 067944 531394 097887 268543 089042 360394 335370 724336 356607 025559 076661 064445 584711 947023 441918 768539 598126 892946 883240 151188 967589 443633 135516 473340 683869 218237 260362 483986 792131 782135 235126 794110 188847 654640 818371 370464 198563 187172 630443 652128 371170 262973 517816 448419 608435 299844 952893 653131 323091 463167 979756 467074 498874 107433 786721 501301 545749 240556 191766 737904 628261 793440 566263 497206 587932 278441 807917 828793 716394 878833 659188 767704 269810 029517 893344 153131 508857 109900 812015 942313 402474 907971 802658 570769 976071 373842 189234 369922 690212 439706 780008 852084 104845 323248 531255 580429 325703 532591 790451 508519 641756 123272 030558 801482 262812 833979 708229 215139 194350 324924 502771 621568 854691 800213 561817 611435 042742 160446 403806 246719 547465 366906 540633 126335 971048 992712 391976 218792 945052 390908 687396 793134 941712 887244 948261 978593 168915 684253 853300 390718 876454 926337 614820 775159 124868 643085 220916 732799 485870 438683 656980 830778 086521 193015 963346 339492 292246 232604 918292 157837 891868 406016 284904 401178 219946 727200 264235 022033 495355 879799 884421 959769 506987 148687 399927 425615 516177 411589 277044 522252 103210 360925 413809 927995 645005 050902 666060 983660 166596 794599 867539 983758 289920 754656 505672 220668 395872 074309 104537 810663 216282 588574 327125 257831 145296 447567 369944 817304 653395 002119 180467 502672 245237 987676 393133 606606 464969 852520 904422 329451 806287 524032 282370 360002 575406 687178 810378 530432 450540 280716 017529 088280 647687 475049 135342 185945 850123 482954 340779 833525 945900 708507 534259 767747 041220 874945 443853 926250 817057 361503 115954 229289 341233 749106 805177 896158 200274 693902 792642 460765 757458 477679 897985 213649 395158 159147 189954 057088 850669 624055 217948 929054 626942 587761 / 1891 > 71997 [i]
- extracting embedded OOA [i] would yield OOA(71997, 667, S7, 3, 1890), but