MinT - Best Known (189−85, 189, s)-Nets in Base 3

Best Known (189−85, 189, s)-Nets in Base 3

(189−85, 189, 80)-Net over F₃ — Constructive and digital

Digital (104, 189, 80)-net over F₃, using

3 times m-reduction [i] based on digital (104, 192, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 96, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(189−85, 189, 128)-Net over F₃ — Digital

Digital (104, 189, 128)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−85, 189, 1087)-Net in Base 3 — Upper bound on s

There is no (104, 189, 1088)-net in base 3, because

1 times m-reduction [i] would yield (104, 188, 1088)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 509005 797769 200922 307791 576362 697276 916195 175447 996196 826121 438694 926373 457237 245832 650113 > 3¹⁸⁸ [i]