MinT - Best Known (158, ∞, s)-Nets in Base 3

Best Known (158, ∞, s)-Nets in Base 3

Digital (158, m, 81)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (158, 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 (158, m, 163)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (158, 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 (158, m, 331)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (158, 1648, 331)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁶⁴⁸, 331, S₃, 5, 1490), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1179 782932 603962 272671 867510 976587 845821 966658 259659 771737 128095 863471 307744 850890 908560 980999 538121 301450 383984 352320 609130 894182 297301 528517 872931 911954 430891 578131 470013 613062 785046 587763 606553 387884 796342 494411 829120 697879 846088 344830 878716 545204 130430 925967 840411 454229 398554 108198 219295 537962 208252 472495 657343 119356 149976 422199 775847 211432 277173 943175 521486 062626 220425 512171 912551 256533 720246 194131 574662 129372 887447 793789 539106 572382 141192 256483 277169 675622 272584 539339 246516 743431 366286 354553 355124 252188 167647 157816 918252 702002 723164 402316 282666 065775 060423 089478 687815 463064 937412 323840 324780 156725 118211 635566 789358 678430 107548 946609 990954 085987 391019 657059 481720 886092 725175 832678 892516 040537 088038 500292 311360 914497 085764 027190 331959 320638 370397 006681 351727 993582 435122 410517 / 497 > 3¹⁶⁴⁸ [i]