MinT - Best Known (157, ∞, s)-Nets in Base 3

Best Known (157, ∞, s)-Nets in Base 3

Digital (157, m, 81)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (157, 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 (157, m, 163)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (157, 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 (157, m, 329)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (157, 1638, 329)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁶³⁸, 329, S₃, 5, 1481), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 10040 064099 766578 941821 399423 548229 775458 323117 047240 399674 268443 467752 041932 050903 883172 922305 460487 601604 220737 485805 027533 084330 501952 841293 913315 940014 650055 797552 599094 415861 463208 502704 265909 369407 932671 659980 107830 536787 360129 959616 017023 466875 695271 952982 753284 911146 517819 228730 310901 302922 751511 562926 125684 386948 119115 299096 847913 902969 071286 960088 946536 801147 606867 953987 898741 591356 333277 712476 203964 575310 015911 270603 377661 328898 392896 390054 364777 839495 301913 378466 163709 395521 752493 560227 385673 070497 091183 004532 376294 093907 046148 814797 306708 871416 477322 083317 805095 717413 613924 112938 183952 299943 367122 965730 355095 418412 895898 709878 125696 697916 507801 561444 725710 604307 121951 887831 616175 345193 579869 918886 983146 136762 541934 785192 299277 898394 497483 664502 394170 326496 026700 / 247 > 3¹⁶³⁸ [i]