MinT - Best Known (136, 136+63, s)-Nets in Base 3

Best Known (136, 136+63, s)-Nets in Base 3

(136, 136+63, 164)-Net over F₃ — Constructive and digital

Digital (136, 199, 164)-net over F₃, using

3¹ times duplication [i] based on digital (135, 198, 164)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 38, 16)-net over F₃, using
    - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
        an explicitly constructive algebraic function field [i]
  2. digital (97, 160, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 80, 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]

(136, 136+63, 373)-Net over F₃ — Digital

Digital (136, 199, 373)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+63, 6894)-Net in Base 3 — Upper bound on s

There is no (136, 199, 6895)-net in base 3, because

1 times m-reduction [i] would yield (136, 198, 6895)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29565 442744 834997 122344 680314 421351 647420 858121 214293 246293 936891 225498 624736 388045 158650 045827 > 3¹⁹⁸ [i]