MinT - Best Known (239−78, 239, s)-Nets in Base 3

Best Known (239−78, 239, s)-Nets in Base 3

(239−78, 239, 167)-Net over F₃ — Constructive and digital

Digital (161, 239, 167)-net over F₃, using

1 times m-reduction [i] based on digital (161, 240, 167)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 48, 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 (113, 192, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 96, 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]

(239−78, 239, 397)-Net over F₃ — Digital

Digital (161, 239, 397)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−78, 239, 6422)-Net in Base 3 — Upper bound on s

There is no (161, 239, 6423)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 079779 438004 554370 075951 184562 063412 168129 184353 957933 708130 960568 172099 111654 554465 005455 489329 762105 660621 033875 > 3²³⁹ [i]