MinT - Best Known (246−159, 246, s)-Nets in Base 3

Best Known (246−159, 246, s)-Nets in Base 3

Digital (87, 246, 62)-net over F₃, using

net from sequence [i] based on digital (87, 61)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 61)-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 (87, 246, 84)-net over F₃, using

There is no digital (87, 246, 304)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁶, 304, F₃, 159) (dual of [304, 58, 160]-code), but
- residual code [i] would yield OA(3⁸⁷, 144, S₃, 53), but
  - the linear programming bound shows that MÂ â‰¥ 17223 084020 715556 855848 832250 872039 681441 493489 720994 373819 071391 134556 757302 805695 961783 787857 312694 961703 747535 992711 367021 / 52541 482507 385915 572483 476316 381490 546694 054744 799929 170031 348199 163698 205504 000000 > 3⁸⁷ [i]

There is no (87, 246, 383)-net in base 3, because

1 times m-reduction [i] would yield (87, 245, 383)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 821 523301 150045 298097 861584 733447 427444 610097 712350 412196 140538 901489 648074 951621 302740 853123 167553 342774 261381 465443 > 3²⁴⁵ [i]