MinT - Best Known (230−115, 230, s)-Nets in Base 3

Best Known (230−115, 230, s)-Nets in Base 3

(230−115, 230, 74)-Net over F₃ — Constructive and digital

Digital (115, 230, 74)-net over F₃, using

t-expansion [i] based on digital (107, 230, 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]

(230−115, 230, 120)-Net over F₃ — Digital

Digital (115, 230, 120)-net over F₃, using

t-expansion [i] based on digital (113, 230, 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]

(230−115, 230, 856)-Net in Base 3 — Upper bound on s

There is no (115, 230, 857)-net in base 3, because

1 times m-reduction [i] would yield (115, 229, 857)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 753690 289500 169326 177715 440124 231145 565057 818983 505759 832712 554243 690773 714693 821815 008295 245956 786980 355155 > 3²²⁹ [i]