MinT - Best Known (211−67, 211, s)-Nets in Base 3

Best Known (211−67, 211, s)-Nets in Base 3

(211−67, 211, 167)-Net over F₃ — Constructive and digital

Digital (144, 211, 167)-net over F₃, using

3¹ times duplication [i] based on digital (143, 210, 167)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 42, 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 (101, 168, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 84, 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]

(211−67, 211, 389)-Net over F₃ — Digital

Digital (144, 211, 389)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−67, 211, 7121)-Net in Base 3 — Upper bound on s

There is no (144, 211, 7122)-net in base 3, because

1 times m-reduction [i] would yield (144, 210, 7122)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 15714 386249 283561 085309 229275 655353 614697 093320 246640 981159 901219 041723 182582 828826 089227 709583 913061 > 3²¹⁰ [i]