MinT - Best Known (109, 109+137, s)-Nets in Base 3

Best Known (109, 109+137, s)-Nets in Base 3

(109, 109+137, 74)-Net over F₃ — Constructive and digital

Digital (109, 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]

(109, 109+137, 104)-Net over F₃ — Digital

Digital (109, 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]

(109, 109+137, 619)-Net in Base 3 — Upper bound on s

There is no (109, 246, 620)-net in base 3, because

1 times m-reduction [i] would yield (109, 245, 620)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 788 758540 816443 902987 858922 400339 362101 362493 294311 279813 586105 174453 211147 795863 048306 308300 551066 035887 360192 340865 > 3²⁴⁵ [i]