MinT - Best Known (219−104, 219, s)-Nets in Base 3

Best Known (219−104, 219, s)-Nets in Base 3

(219−104, 219, 76)-Net over F₃ — Constructive and digital

Digital (115, 219, 76)-net over F₃, using

6 times m-reduction [i] based on digital (115, 225, 76)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 70, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 155, 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]

(219−104, 219, 125)-Net over F₃ — Digital

Digital (115, 219, 125)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−104, 219, 982)-Net in Base 3 — Upper bound on s

There is no (115, 219, 983)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 309 132893 000876 075039 536587 344923 703581 977656 578732 301180 390853 018357 054910 476964 119714 462250 300863 663897 > 3²¹⁹ [i]