MinT - Best Known (236−125, 236, s)-Nets in Base 3

Best Known (236−125, 236, s)-Nets in Base 3

(236−125, 236, 74)-Net over F₃ — Constructive and digital

Digital (111, 236, 74)-net over F₃, using

t-expansion [i] based on digital (107, 236, 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]

(236−125, 236, 104)-Net over F₃ — Digital

Digital (111, 236, 104)-net over F₃, using

t-expansion [i] based on digital (102, 236, 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]

(236−125, 236, 710)-Net in Base 3 — Upper bound on s

There is no (111, 236, 711)-net in base 3, because

1 times m-reduction [i] would yield (111, 235, 711)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14036 900215 818286 331799 050350 042945 088775 039332 322840 341717 080431 486354 851426 899954 683296 342912 907637 165297 312981 > 3²³⁵ [i]