MinT - Best Known (233−127, 233, s)-Nets in Base 3

Best Known (233−127, 233, s)-Nets in Base 3

(233−127, 233, 73)-Net over F₃ — Constructive and digital

Digital (106, 233, 73)-net over F₃, using

net from sequence [i] based on digital (106, 72)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 72)-sequence over F₉, using
  - s-reduction 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]

(233−127, 233, 104)-Net over F₃ — Digital

Digital (106, 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]

(233−127, 233, 634)-Net in Base 3 — Upper bound on s

There is no (106, 233, 635)-net in base 3, because

1 times m-reduction [i] would yield (106, 232, 635)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 528 774260 202567 890598 330391 916725 486916 511223 737207 061114 907255 367242 266665 332415 247521 740641 150422 713999 164019 > 3²³² [i]