MinT - Best Known (227−103, 227, s)-Nets in Base 3

Best Known (227−103, 227, s)-Nets in Base 3

(227−103, 227, 85)-Net over F₃ — Constructive and digital

Digital (124, 227, 85)-net over F₃, using

1 times m-reduction [i] based on digital (124, 228, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 79, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
  2. digital (45, 149, 48)-net over F₃, using
    - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
        an explicitly constructive algebraic function field [i]

(227−103, 227, 147)-Net over F₃ — Digital

Digital (124, 227, 147)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−103, 227, 1241)-Net in Base 3 — Upper bound on s

There is no (124, 227, 1242)-net in base 3, because

1 times m-reduction [i] would yield (124, 226, 1242)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 677678 809324 632874 847550 172556 199773 816025 011703 893958 480813 613202 011349 642171 739578 494945 012580 105105 853585 > 3²²⁶ [i]