MinT - Best Known (153, ∞, s)-Nets in Base 3

Best Known (153, ∞, s)-Nets in Base 3

Digital (153, m, 81)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (153, 80)-sequence over F₃, using
- t-expansion [i] based on digital (144, 80)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

Digital (153, m, 163)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (153, 162)-sequence over F₃, using
- t-expansion [i] based on digital (151, 162)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 151 and N(F) ≥ 163, using
    - Drinfeld modules of rank 1 [i]

There is no (153, m, 321)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (153, 1598, 321)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁵⁹⁸, 321, S₃, 5, 1445), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 842 338145 941737 022093 922489 753061 468430 302940 112439 288104 314816 915874 381547 015446 261486 760183 390511 207767 009443 782416 170947 670765 285607 779258 217661 804916 089867 326570 163466 870359 773067 822161 237496 866500 643851 302574 404042 455374 332562 149278 299073 611630 345124 573080 030264 753502 825279 499560 584499 294957 701808 049635 143242 369818 681588 895898 460057 763513 700239 728870 978777 140242 139107 725950 943822 427394 170602 676908 716644 431481 933180 677464 990614 566229 176383 098015 408993 177297 994809 502125 832657 722252 726053 782443 794431 046934 021920 654125 019033 345266 203989 341574 893380 064358 182718 327730 059355 658847 345112 336377 584507 840475 006210 474120 892502 465874 703963 707107 082960 806016 156591 743587 656718 775604 905299 560785 762864 468266 115474 366695 371653 408915 859879 602215 457903 865808 241211 968034 / 241 > 3¹⁵⁹⁸ [i]