MinT - Best Known (86, 241, s)-Nets in Base 3

Best Known (86, 241, s)-Nets in Base 3

Digital (86, 241, 61)-net over F₃, using

net from sequence [i] based on digital (86, 60)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 60)-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 (86, 241, 84)-net over F₃, using

There is no digital (86, 241, 375)-net over F₃, because

2 times m-reduction [i] would yield digital (86, 239, 375)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁹, 375, F₃, 153) (dual of [375, 136, 154]-code), but
  - residual code [i] would yield linear OA(3⁸⁶, 221, F₃, 51) (dual of [221, 135, 52]-code), but
    - 1 times truncation [i] would yield linear OA(3⁸⁵, 220, F₃, 50) (dual of [220, 135, 51]-code), but
      - the Johnson bound shows that NÂ â‰¤ 22905 993227 448745 319194 488151 012424 723463 351500 755461 393112 800141 < 3¹³⁵ [i]

There is no (86, 241, 382)-net in base 3, because

1 times m-reduction [i] would yield (86, 240, 382)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 796581 915971 615608 596523 149700 544706 311875 331184 992212 526237 337567 117588 488786 634024 713853 923480 188525 480032 168837 > 3²⁴⁰ [i]