MinT - Best Known (240−121, 240, s)-Nets in Base 3

Best Known (240−121, 240, s)-Nets in Base 3

(240−121, 240, 76)-Net over F₃ — Constructive and digital

Digital (119, 240, 76)-net over F₃, using

net from sequence [i] based on digital (119, 75)-sequence over F₃, using
- base reduction for sequences [i] based on digital (22, 75)-sequence over F₉, using
  - s-reduction based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(240−121, 240, 120)-Net over F₃ — Digital

Digital (119, 240, 120)-net over F₃, using

t-expansion [i] based on digital (113, 240, 120)-net over F₃, using
- net from sequence [i] based on digital (113, 119)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(240−121, 240, 864)-Net in Base 3 — Upper bound on s

There is no (119, 240, 865)-net in base 3, because

1 times m-reduction [i] would yield (119, 239, 865)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 128445 898060 301510 079214 478874 642578 591397 408248 143055 788598 723649 909526 300220 917922 083057 375917 971849 118979 957937 > 3²³⁹ [i]