MinT - Best Known (168−82, 168, s)-Nets in Base 3

Best Known (168−82, 168, s)-Nets in Base 3

(168−82, 168, 65)-Net over F₃ — Constructive and digital

Digital (86, 168, 65)-net over F₃, using

6 times m-reduction [i] based on digital (86, 174, 65)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 59, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (27, 115, 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]

(168−82, 168, 92)-Net over F₃ — Digital

Digital (86, 168, 92)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−82, 168, 688)-Net in Base 3 — Upper bound on s

There is no (86, 168, 689)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 151 207449 908090 113910 130374 704272 381296 637821 292519 318842 740208 693849 609550 993603 > 3¹⁶⁸ [i]