MinT - Best Known (145, 145+69, s)-Nets in Base 3

Best Known (145, 145+69, s)-Nets in Base 3

(145, 145+69, 164)-Net over F₃ — Constructive and digital

Digital (145, 214, 164)-net over F₃, using

3¹ times duplication [i] based on digital (144, 213, 164)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 41, 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 (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]

(145, 145+69, 375)-Net over F₃ — Digital

Digital (145, 214, 375)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(145, 145+69, 6565)-Net in Base 3 — Upper bound on s

There is no (145, 214, 6566)-net in base 3, because

1 times m-reduction [i] would yield (145, 213, 6566)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 424689 838510 520432 813982 351904 656336 990929 616499 709143 426273 745287 037058 422530 991469 334273 951254 351213 > 3²¹³ [i]