MinT - Best Known (75, 75+136, s)-Nets in Base 3

Best Known (75, 75+136, s)-Nets in Base 3

Digital (75, 211, 50)-net over F₃, using

net from sequence [i] based on digital (75, 49)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 49)-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 (75, 211, 84)-net over F₃, using

There is no digital (75, 211, 284)-net over F₃, because

1 times m-reduction [i] would yield digital (75, 210, 284)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²¹⁰, 284, F₃, 135) (dual of [284, 74, 136]-code), but
  - residual code [i] would yield OA(3⁷⁵, 148, S₃, 45), but
    - the linear programming bound shows that MÂ â‰¥ 590 120846 623377 656391 069611 419462 478892 700111 332235 753542 892978 964081 186500 581837 397525 439282 627472 097286 950369 041256 226115 254166 206219 748203 859328 746017 067633 491684 819772 271401 346832 059505 067379 993239 767275 842410 377928 173211 258457 224366 465382 215796 051686 653704 753388 169641 277219 669316 425315 179848 202413 868104 710118 413044 518919 058715 102177 180355 338024 292294 755043 575759 / 875 283163 572176 128193 747686 199754 076313 595617 360350 823278 793844 480930 816971 707313 086974 061443 650139 220826 606324 078439 870380 390239 549832 592829 508396 019427 758043 098233 935504 073091 399672 564689 457449 221176 207597 355744 849697 870861 217835 478980 011951 210294 662331 675715 140506 357595 012793 982579 054478 875395 189571 057565 271813 852959 656730 623375 > 3⁷⁵ [i]

There is no (75, 211, 298)-net in base 3, because

27 times m-reduction [i] would yield (75, 184, 298)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁸⁴, 298, S₃, 109), but
  - 2 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]