MinT - Best Known (168, 168+79, s)-Nets in Base 3

Best Known (168, 168+79, s)-Nets in Base 3

(168, 168+79, 176)-Net over F₃ — Constructive and digital

Digital (168, 247, 176)-net over F₃, using

3¹ times duplication [i] based on digital (167, 246, 176)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 54, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, 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]

(168, 168+79, 435)-Net over F₃ — Digital

Digital (168, 247, 435)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(168, 168+79, 7830)-Net in Base 3 — Upper bound on s

There is no (168, 247, 7831)-net in base 3, because

1 times m-reduction [i] would yield (168, 246, 7831)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2356 971635 813058 622986 451893 138937 912160 909138 674449 107119 579816 876005 281383 507935 448973 673889 710685 387514 677830 750099 > 3²⁴⁶ [i]