MinT - Best Known (165, 165+81, s)-Nets in Base 3

Best Known (165, 165+81, s)-Nets in Base 3

(165, 165+81, 167)-Net over F₃ — Constructive and digital

Digital (165, 246, 167)-net over F₃, using

3¹ times duplication [i] based on digital (164, 245, 167)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 49, 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 (115, 196, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 98, 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]

(165, 165+81, 396)-Net over F₃ — Digital

Digital (165, 246, 396)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+81, 6554)-Net in Base 3 — Upper bound on s

There is no (165, 246, 6555)-net in base 3, because

1 times m-reduction [i] would yield (165, 245, 6555)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 788 079891 591946 558752 536833 535816 803578 555667 527413 809267 790719 225214 866761 129682 445509 706331 882157 574124 520294 634289 > 3²⁴⁵ [i]