MinT - Best Known (111, 111+135, s)-Nets in Base 3

Best Known (111, 111+135, s)-Nets in Base 3

(111, 111+135, 74)-Net over F₃ — Constructive and digital

Digital (111, 246, 74)-net over F₃, using

t-expansion [i] based on digital (107, 246, 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]

(111, 111+135, 104)-Net over F₃ — Digital

Digital (111, 246, 104)-net over F₃, using

t-expansion [i] based on digital (102, 246, 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]

(111, 111+135, 652)-Net in Base 3 — Upper bound on s

There is no (111, 246, 653)-net in base 3, because

1 times m-reduction [i] would yield (111, 245, 653)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 849 303623 734704 889722 469869 214057 125155 475799 677328 975721 876960 360488 905866 471886 647985 009078 914848 275597 209080 950219 > 3²⁴⁵ [i]