MinT - Best Known (216−69, 216, s)-Nets in Base 3

Best Known (216−69, 216, s)-Nets in Base 3

(216−69, 216, 167)-Net over F₃ — Constructive and digital

Digital (147, 216, 167)-net over F₃, using

3¹ times duplication [i] based on digital (146, 215, 167)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 43, 19)-net over F₃, using
    - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [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 (103, 172, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 86, 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]

(216−69, 216, 389)-Net over F₃ — Digital

Digital (147, 216, 389)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−69, 216, 7005)-Net in Base 3 — Upper bound on s

There is no (147, 216, 7006)-net in base 3, because

1 times m-reduction [i] would yield (147, 215, 7006)-net in base 3, but
- 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]