MinT - Best Known (87, 241, s)-Nets in Base 3

Best Known (87, 241, s)-Nets in Base 3

Digital (87, 241, 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, 241, 84)-net over F₃, using

There is no digital (87, 241, 384)-net over F₃, because

1 times m-reduction [i] would yield digital (87, 240, 384)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁰, 384, F₃, 153) (dual of [384, 144, 154]-code), but
  - residual code [i] would yield linear OA(3⁸⁷, 230, F₃, 51) (dual of [230, 143, 52]-code), but
    - 1 times truncation [i] would yield linear OA(3⁸⁶, 229, F₃, 50) (dual of [229, 143, 51]-code), but
      - the Johnson bound shows that NÂ â‰¤ 153 153734 778487 159186 543991 832667 435898 157821 534223 513753 575995 914628 < 3¹⁴³ [i]

There is no (87, 241, 388)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 548795 724917 665078 015330 713411 563719 848244 448696 089036 441285 767779 802534 685907 262067 325852 624630 599619 141837 786729 > 3²⁴¹ [i]