MinT - Best Known (234−112, 234, s)-Nets in Base 3

Best Known (234−112, 234, s)-Nets in Base 3

(234−112, 234, 80)-Net over F₃ — Constructive and digital

Digital (122, 234, 80)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 77, 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 (45, 157, 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]

(234−112, 234, 130)-Net over F₃ — Digital

Digital (122, 234, 130)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−112, 234, 1015)-Net in Base 3 — Upper bound on s

There is no (122, 234, 1016)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4559 098106 990931 607965 601343 478616 743123 462035 752448 908932 395813 206990 080628 245182 116412 613659 219806 074767 236353 > 3²³⁴ [i]