MinT - Best Known (233−143, 233, s)-Nets in Base 3

Best Known (233−143, 233, s)-Nets in Base 3

(233−143, 233, 64)-Net over F₃ — Constructive and digital

Digital (90, 233, 64)-net over F₃, using

t-expansion [i] based on digital (89, 233, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(233−143, 233, 96)-Net over F₃ — Digital

Digital (90, 233, 96)-net over F₃, using

t-expansion [i] based on digital (89, 233, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(233−143, 233, 427)-Net in Base 3 — Upper bound on s

There is no (90, 233, 428)-net in base 3, because

1 times m-reduction [i] would yield (90, 232, 428)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 545 052647 576416 147004 712070 143058 062348 361778 853941 853931 325931 768136 773477 563045 254300 438791 725353 414835 772369 > 3²³² [i]