MinT - Best Known (233−123, 233, s)-Nets in Base 3

Best Known (233−123, 233, s)-Nets in Base 3

(233−123, 233, 74)-Net over F₃ — Constructive and digital

Digital (110, 233, 74)-net over F₃, using

t-expansion [i] based on digital (107, 233, 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]

(233−123, 233, 104)-Net over F₃ — Digital

Digital (110, 233, 104)-net over F₃, using

t-expansion [i] based on digital (102, 233, 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]

(233−123, 233, 710)-Net in Base 3 — Upper bound on s

There is no (110, 233, 711)-net in base 3, because

1 times m-reduction [i] would yield (110, 232, 711)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 524 187365 250449 303905 969314 092928 317843 009717 655771 055119 491024 977874 644582 261356 582936 718832 706694 512644 070487 > 3²³² [i]