MinT - Best Known (168−78, 168, s)-Nets in Base 3

Best Known (168−78, 168, s)-Nets in Base 3

(168−78, 168, 69)-Net over F₃ — Constructive and digital

Digital (90, 168, 69)-net over F₃, using

6 times m-reduction [i] based on digital (90, 174, 69)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 63, 32)-net over F₃, using
    - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
        an explicitly constructive algebraic function field [i]
  2. digital (27, 111, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(168−78, 168, 106)-Net over F₃ — Digital

Digital (90, 168, 106)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−78, 168, 836)-Net in Base 3 — Upper bound on s

There is no (90, 168, 837)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 146 288717 068747 676862 385587 865042 559423 718543 564875 976346 199426 998982 426585 842011 > 3¹⁶⁸ [i]