MinT - Best Known (135, ∞, s)-Nets in Base 3

Best Known (135, ∞, s)-Nets in Base 3

Digital (135, m, 78)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (135, 77)-sequence over F₃, using
- t-expansion [i] based on digital (121, 77)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

Digital (135, m, 120)-net over F₃ for arbitrarily large m, using

There is no (135, m, 284)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (135, 1697, 284)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁶⁹⁷, 284, S₃, 6, 1562), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 277 592265 006008 785121 138278 940554 606758 653960 950297 820520 229524 893645 737613 231224 398567 676865 031726 029838 219806 379402 398496 457106 185058 315287 535926 828701 004784 487431 785027 217309 872435 393999 731628 847424 728023 754465 409905 454508 313485 104256 541008 596137 344791 413006 990034 006644 553603 072014 927044 538281 272798 594184 475745 155842 314165 104335 803042 282955 657252 187159 173819 122823 769822 920533 810641 789841 546624 165580 962776 601094 407192 967227 116313 757122 304451 841624 047617 552128 601234 892511 629228 794263 544577 495993 141018 126020 518883 078472 148744 948751 626522 085058 954799 559041 937088 999318 874145 161886 313462 805801 535657 012187 399866 492004 504015 345227 211383 039434 158691 514747 625246 599248 633790 365048 790815 806936 669762 769333 761053 815386 711410 691561 259536 662690 272125 753474 394164 281979 884105 045045 646135 659709 459363 183604 444801 853281 / 521 > 3¹⁶⁹⁷ [i]