MinT - Best Known (215−68, 215, s)-Nets in Base 3

Best Known (215−68, 215, s)-Nets in Base 3

(215−68, 215, 168)-Net over F₃ — Constructive and digital

Digital (147, 215, 168)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 45, 20)-net over F₃, using
  - net from sequence [i] based on digital (11, 19)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 9, N(F)Â =Â 19, and 1 place with degree 3 [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]
2. digital (102, 170, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 85, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(215−68, 215, 400)-Net over F₃ — Digital

Digital (147, 215, 400)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−68, 215, 7005)-Net in Base 3 — Upper bound on s

There is no (147, 215, 7006)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 811865 452682 087882 904863 972238 809033 099147 916708 175071 345670 310304 351159 723213 979515 544350 600058 681725 > 3²¹⁵ [i]