MinT - Best Known (218−71, 218, s)-Nets in Base 3

Best Known (218−71, 218, s)-Nets in Base 3

(218−71, 218, 164)-Net over F₃ — Constructive and digital

Digital (147, 218, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 42, 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 (105, 176, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 88, 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]

(218−71, 218, 369)-Net over F₃ — Digital

Digital (147, 218, 369)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−71, 218, 6280)-Net in Base 3 — Upper bound on s

There is no (147, 218, 6281)-net in base 3, because

1 times m-reduction [i] would yield (147, 217, 6281)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 347819 041815 654775 671982 649806 430376 838952 593492 667957 981684 266991 897636 674496 845078 629072 279561 898235 > 3²¹⁷ [i]