MinT - Best Known (205−104, 205, s)-Nets in Base 3

Best Known (205−104, 205, s)-Nets in Base 3

(205−104, 205, 69)-Net over F₃ — Constructive and digital

Digital (101, 205, 69)-net over F₃, using

2 times m-reduction [i] based on digital (101, 207, 69)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 74, 32)-net over F₃, using
    - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
        an explicitly constructive algebraic function field [i]
  2. digital (27, 133, 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]

(205−104, 205, 98)-Net over F₃ — Digital

Digital (101, 205, 98)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(205−104, 205, 718)-Net in Base 3 — Upper bound on s

There is no (101, 205, 719)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 65 826706 778285 534125 787716 914833 668144 607484 289290 682386 137859 734886 160598 719931 285136 847640 008089 > 3²⁰⁵ [i]