MinT - Best Known (74, 74+135, s)-Nets in Base 3

Best Known (74, 74+135, s)-Nets in Base 3

Digital (74, 209, 49)-net over F₃, using

net from sequence [i] based on digital (74, 48)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 48)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

Digital (74, 209, 84)-net over F₃, using

There is no digital (74, 209, 271)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁰⁹, 271, F₃, 135) (dual of [271, 62, 136]-code), but
- residual code [i] would yield OA(3⁷⁴, 135, S₃, 45), but
  - the linear programming bound shows that MÂ â‰¥ 3149 102978 502159 702832 324850 000724 343690 781397 590391 301020 103565 627253 799681 342733 232096 935540 172381 288036 145917 424600 949572 865273 314090 681217 849098 897411 221037 317415 036463 486694 956203 154705 978691 418966 997838 451810 319831 790028 969021 194326 831692 990368 995587 851882 061731 / 14597 778493 990865 194663 196016 206641 110192 731698 087051 433957 071155 012610 160359 950448 727704 848834 991243 819836 175998 265310 906935 374233 433468 848031 172344 125058 682610 464451 658601 633589 335287 574794 198669 581670 459179 939879 334859 216869 016355 > 3⁷⁴ [i]

There is no (74, 209, 297)-net in base 3, because

26 times m-reduction [i] would yield (74, 183, 297)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁸³, 297, S₃, 109), but
  - 3 times code embedding in larger space [i] would yield OA(3¹⁸⁶, 300, S₃, 109), but
    - the linear programming bound shows that MÂ â‰¥ 744 646766 850603 526568 601268 797422 932026 535817 992617 489413 629925 481692 835741 945421 210090 969950 381274 252816 504020 353660 158683 457200 925516 493661 440389 831805 338854 508258 811020 990136 433923 762970 235494 388055 889637 880458 633459 997426 817105 857222 614437 226587 / 10512 885684 310613 723066 595609 767913 116130 406095 478720 857818 594175 294259 025889 504412 156455 324830 169263 105843 888569 812652 562285 873819 443701 107860 316189 843301 339995 > 3¹⁸⁶ [i]