MinT - Best Known (172−82, 172, s)-Nets in Base 3

Best Known (172−82, 172, s)-Nets in Base 3

(172−82, 172, 69)-Net over F₃ — Constructive and digital

Digital (90, 172, 69)-net over F₃, using

2 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]

(172−82, 172, 101)-Net over F₃ — Digital

Digital (90, 172, 101)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−82, 172, 770)-Net in Base 3 — Upper bound on s

There is no (90, 172, 771)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 12038 981709 280670 320393 940084 441108 406298 146258 997632 637175 822140 571965 186418 442359 > 3¹⁷² [i]