MinT - Best Known (220−98, 220, s)-Nets in Base 3

Best Known (220−98, 220, s)-Nets in Base 3

(220−98, 220, 85)-Net over F₃ — Constructive and digital

Digital (122, 220, 85)-net over F₃, using

2 times m-reduction [i] based on digital (122, 222, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 77, 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, 145, 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]

(220−98, 220, 150)-Net over F₃ — Digital

Digital (122, 220, 150)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(220−98, 220, 1277)-Net in Base 3 — Upper bound on s

There is no (122, 220, 1278)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 927 636749 747551 889809 776085 393466 448822 591222 271967 541514 610341 874970 784865 779259 527249 564605 118203 485085 > 3²²⁰ [i]