MinT - Best Known (218−69, 218, s)-Nets in Base 3

Best Known (218−69, 218, s)-Nets in Base 3

(218−69, 218, 168)-Net over F₃ — Constructive and digital

Digital (149, 218, 168)-net over F₃, using

3¹ times duplication [i] based on digital (148, 217, 168)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 45, 20)-net over F₃, using
    - net from sequence [i] based on digital (11, 19)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 9, N(F)Â =Â 19, and 1 place with degree 3 [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 (103, 172, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 86, 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]

(218−69, 218, 404)-Net over F₃ — Digital

Digital (149, 218, 404)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−69, 218, 7475)-Net in Base 3 — Upper bound on s

There is no (149, 218, 7476)-net in base 3, because

1 times m-reduction [i] would yield (149, 217, 7476)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 316858 371739 312229 771255 786487 486388 865604 026335 640197 116555 539032 361067 096492 431987 711292 725377 634393 > 3²¹⁷ [i]