MinT - Best Known (238−127, 238, s)-Nets in Base 3

Best Known (238−127, 238, s)-Nets in Base 3

(238−127, 238, 74)-Net over F₃ — Constructive and digital

Digital (111, 238, 74)-net over F₃, using

t-expansion [i] based on digital (107, 238, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(238−127, 238, 104)-Net over F₃ — Digital

Digital (111, 238, 104)-net over F₃, using

t-expansion [i] based on digital (102, 238, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(238−127, 238, 697)-Net in Base 3 — Upper bound on s

There is no (111, 238, 698)-net in base 3, because

1 times m-reduction [i] would yield (111, 237, 698)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 126743 535431 531735 261443 312808 061134 190382 021243 728998 564027 471144 193483 229289 233299 319414 431378 240162 228809 582937 > 3²³⁷ [i]