MinT - Best Known (232−149, 232, s)-Nets in Base 3

Best Known (232−149, 232, s)-Nets in Base 3

Digital (83, 232, 58)-net over F₃, using

net from sequence [i] based on digital (83, 57)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 57)-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 (83, 232, 84)-net over F₃, using

There is no digital (83, 232, 364)-net over F₃, because

2 times m-reduction [i] would yield digital (83, 230, 364)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁰, 364, F₃, 147) (dual of [364, 134, 148]-code), but
  - residual code [i] would yield linear OA(3⁸³, 216, F₃, 49) (dual of [216, 133, 50]-code), but
    - 1 times truncation [i] would yield linear OA(3⁸², 215, F₃, 48) (dual of [215, 133, 49]-code), but
      - the Johnson bound shows that NÂ â‰¤ 2633 391490 584373 459265 026071 777086 821369 449862 527703 301746 510338 < 3¹³³ [i]

There is no (83, 232, 370)-net in base 3, because

1 times m-reduction [i] would yield (83, 231, 370)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 192 410633 408040 302953 516525 217476 562977 107486 350283 844103 677088 086139 494183 984364 213012 746341 778493 115458 691333 > 3²³¹ [i]