MinT - Best Known (165, 165+78, s)-Nets in Base 3

Best Known (165, 165+78, s)-Nets in Base 3

(165, 165+78, 172)-Net over F₃ — Constructive and digital

Digital (165, 243, 172)-net over F₃, using

1 times m-reduction [i] based on digital (165, 244, 172)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 52, 24)-net over F₃, using
    - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
        a function field by SÃ©mirat [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]

(165, 165+78, 424)-Net over F₃ — Digital

Digital (165, 243, 424)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+78, 7193)-Net in Base 3 — Upper bound on s

There is no (165, 243, 7194)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 87 626235 532120 289897 705871 247086 828339 073291 520966 325263 250675 714376 833584 284207 754002 709958 237547 783577 998380 931017 > 3²⁴³ [i]