MinT - Best Known (232−147, 232, s)-Nets in Base 3

Best Known (232−147, 232, s)-Nets in Base 3

Digital (85, 232, 60)-net over F₃, using

net from sequence [i] based on digital (85, 59)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 59)-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 (85, 232, 84)-net over F₃, using

There is no digital (85, 232, 384)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²³², 384, F₃, 147) (dual of [384, 152, 148]-code), but
- residual code [i] would yield linear OA(3⁸⁵, 236, F₃, 49) (dual of [236, 151, 50]-code), but
  - 1 times truncation [i] would yield linear OA(3⁸⁴, 235, F₃, 48) (dual of [235, 151, 49]-code), but
    - the Johnson bound shows that NÂ â‰¤ 1 014891 606187 829584 057582 115648 929135 604017 524062 355605 507975 062077 667701 < 3¹⁵¹ [i]

There is no (85, 232, 385)-net in base 3, because

1 times m-reduction [i] would yield (85, 231, 385)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 167 388465 647247 039735 721874 154753 702320 172230 921427 532388 266684 459967 902510 106459 511350 210254 742537 566687 592163 > 3²³¹ [i]